670:
1168:
1602:
1415:
355:
31:
2979:
1392:
First, we link all the immediate children nodes of a given parent node together in order to form a link list. Then, we keep the link from the parent to the first (i.e., the leftmost) child and remove all the other links to the rest of the children. We repeat this process for all the children (if they
1712:
The problem with this representation is that listing all bit strings in lexicographic order would mean two successive strings might represent two trees that are lexicographically very different. Therefore, enumeration over binary strings would not necessarily result in an ordered generation of all
3939:
children representing the next possible character in the string. Thus, characters along the paths can represent valid keys by marking the end character of the keys as "terminal node". For example, in the example below "at" and "and" are valid key strings with "t" and "d" marked as terminal nodes.
1037:-ary tree is very similar to traversing a binary tree. The pre-order traversal goes to parent, left subtree and the right subtree, and for traversing post-order it goes by left subtree, right subtree, and parent node. For traversing in-order, since there are more than two children per node for
665:{\displaystyle {\begin{aligned}&{\frac {m^{h+1}-1}{m-1}}\geq N>{\frac {m^{h}-1}{m-1}}\\&m^{h+1}\geq (m-1)\cdot N+1>m^{h}\\&h+1\geq \log _{m}\left((m-1)\cdot N+1\right)>h\\&h\geq \left\lceil \log _{m}((m-1)\cdot N+1)-1\right\rceil .\end{aligned}}}
2416:
Once one has exhausted all possible positions in the backbone template, a new template will be constructed by shifting the 3rd node one position to the right as depicted below, and the same enumeration would occur until all possible positions labeled "X" is exhausted.
2487:
3751:
827:
1387:
3956:
1698:
1960:
2138:
2405:
that governs the generation of the possible ordered trees starting from "000" to "006". The backbone template for this group ("00X") is depicted below, where an additional node is added in the positions labeled "x".
360:
1008:
1877:
343:
1527:
1102:
4067:
1153:
3610:
901:
675:
1258:
3405:
1790:
3108:
1393:
have any children) until we have processed all the internal nodes and rotate the tree by 45 degrees clockwise. The tree obtained is the desired binary tree obtained from the given
3601:
3487:
2011:
1263:
1697:
1646:
1483:
1568:
1190:
ary tree to a binary tree would only increase the height of the tree by a constant factor and would not affect the overall worst-case time complexity. In other words,
2198:
3541:
3303:
3242:
3162:
3135:
208:
2715:
2682:
2044:
3340:
3268:
2893:
2481:
2455:
2168:
3191:
2973:
2953:
2933:
2913:
2867:
2847:
2827:
2807:
2783:
2763:
2743:
2049:
1596:
249:
2144:-ary tree cannot exceed the total number of null pointers (i.e., pointers without any child node attached to them). This summation is putting restriction on
4196:
1886:
2421:
2410:
1049:
subtrees. One common method to establish left/right subtrees is to divide the list of children nodes into two groups. By defining an order on the
4600:
4206:
4113:
4077:
4021:
4807:
3973:
3940:
Terminal nodes can store extra information to be associated with a given key. There are similar ways to building such a dictionary using
917:
1717:
ary trees. A better representation is based on an integer string that indicates the number of zeroes between successive ones, known as
4351:
2483:, we can easily observe the apparent jump from "006" to "010" can be explained trivially in an algorithmic fashion as depicted below:
4305:
4158:
4321:
4868:
4467:
1795:
258:
1489:
1056:
1883:
is the number of zeroes needed at the tail end of the sequence to make the string have the appropriate length. For example,
1107:
4863:
2684:
worst-case time are called loopless since the time complexity cannot involve a loop or recursion. Loopless enumeration of
843:
4181:
4129:
4052:
1186:
children. As a result, this fact leads to a sparse array with large unused space in the memory. Converting an arbitrary
1530:(assuming the root has index zero, meaning a 0-based array). This method benefits from more compact storage and better
78:
4432:
1193:
4817:
2017:, which duplicate bases cannot be adjacent. This new representation allows to construct a next valid sequence in
1661:
3345:
4373:
2200:
without creating an invalid structure (i.e. having an available null pointer to attached the last node to it).
1670:-ary trees is useful in many disciplines as a way of checking hypotheses or theories. Proper representation of
3054:
1724:
3746:{\displaystyle \sum _{i=j}^{n}\sum _{t=2}^{m}c_{(i-1)(m-1)+t-1}\qquad \leq n-j,\qquad \forall j\in 0\dots n.}
1686:
nodes indicating the presence of a node at a given index using binary values. For example, the bit sequence
3935:) with the root of the tree representing the starting point. Similarly, each of the children can have up to
1430:
822:{\displaystyle D=h\geq \left\lceil \log _{m}((m-1)\cdot N+1)-1\right\rceil =O(\log _{m}n)=O(\log n/\log m).}
221:-ary tree does not include the root node, with a tree containing only a root node having a height of 0.
4344:
1382:{\textstyle \log _{2}m\cdot \log _{m}n={\frac {\log m}{\log 2}}\cdot {\frac {\log n}{\log m}}=\log _{2}n}
4511:
4360:
3550:
3426:
1531:
4501:
4456:
4144:
1969:
1434:
1613:
Compared to array-based implementation, this implementation method has superior space complexity of
4615:
4391:
4471:
3927:-ary tree is creating a dictionary for validation of acceptable strings. In order to do that, let
2688:
ary trees is said to be loopless if after initialization, it generates successive tree objects in
4782:
4741:
4567:
4557:
4436:
4175:
4123:
4046:
4027:
1962:
is the simple zero sequence representation of the above figure. A more compact representation of
1616:
1456:
1182:
ary tree is inefficient, because most of the nodes in practical applications contain less than
4676:
4377:
4337:
4301:
4202:
4154:
4109:
4073:
4017:
1537:
4767:
4279:
4249:
4009:
3968:
3932:
3504:-ary tree. Thus, enumerating all possible legal encoding would helps us to generate all the
2420:
2173:
1441:-ary tree, this method wastes no space. In this compact arrangement, if a node has an index
3999:
3519:
3281:
3220:
3140:
3113:
2409:
186:
4711:
4461:
2691:
2658:
2020:
3319:
3247:
2872:
2460:
2434:
2147:
4664:
4659:
4542:
4476:
3167:
2958:
2938:
2918:
2898:
2852:
2832:
2812:
2792:
2768:
2748:
2728:
1581:
1028:
1011:
234:
4857:
4812:
4792:
4635:
4524:
4451:
4283:
4143:
Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2022).
3313:
ary tree that can be recovered by performing the same sequence of right-t rotations.
4386:
1955:{\displaystyle 1110000100010001000\equiv 10^{0}10^{0}10^{4}10^{4}10^{3}\equiv 00433}
1534:, particularly during a preorder traversal. The space complexity of this method is
4772:
4736:
4552:
4547:
4529:
4441:
4240:
Baronaigien, Dominique
Roelants van (2000). "Loop Free Generation of K-ary trees".
3274:
at each depth. Then, the sequence of number of left-t rotations performed at depth
2486:
98:
82:
42:
4031:
1674:
ary tree objects can greatly simplify the generation process. One can construct a
4822:
4787:
4777:
4691:
4625:
4620:
4610:
4519:
4368:
3978:
90:
4072:. Appendix: Graph Theory Terminology: Cambridge University Press. p. 573.
4832:
4802:
4762:
4605:
4534:
4481:
4401:
4013:
3217:. Specifically, this can be done by performing left-t rotations on each node
4837:
4797:
4644:
4572:
4562:
1167:
158:
4253:
1578:
Each node would have an internal array for storing pointers to each of its
4726:
4416:
3931:
be equal to the number of valid alphabets (e.g., number of letters of the
2133:{\displaystyle \sum _{i=1}^{i=j}s_{i}\leq (m-1)j\qquad \forall j\leq n-1.}
17:
4842:
4716:
4696:
4669:
4654:
4406:
3500:
Capturing counts of left-rotations at each depth is a way of encoding an
2203:
The table below shows the list of all valid simple zero sequences of all
1601:
1414:
4827:
4731:
4706:
4649:
4496:
4426:
4421:
4396:
4329:
4198:
The
Autopoiesis of Architecture: A New Agenda for Architecture, Vol. II
3955:
2978:
4270:
Korsh, James F (1994). "Loopless generation of k-ary tree sequences".
3954:
2977:
2485:
2419:
2408:
2401:
Starting from the bottom right of the table (i.e., "000"), there is a
4746:
4721:
4701:
4686:
4595:
4486:
4411:
3945:
3941:
1003:{\textstyle C_{n}={\frac {1}{(m-1)n+1}}\cdot {\binom {m\cdot n}{n}}}
30:
4590:
4491:
4446:
3547:
non-negative integers represent a valid m-ary tree. A sequence of
1600:
1413:
1166:
29:
4582:
3949:
4333:
27:
Tree data structure in which each node has at most m children.
4222:
Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994).
2140:
That is to say that number of zeros in the bit sequence of a
1792:
is a Simple Zero
Sequence corresponding to the bit sequence
1696:
1429:-ary trees can also be stored in breadth-first order as an
1872:{\textstyle 10^{s_{1}}10^{s_{2}}\ldots 10^{s_{n-1}}10^{j}}
1171:
An example of conversion of a m-ary tree to a binary tree.
338:{\textstyle \sum _{i=0}^{h}m^{i}={\frac {m^{h+1}-1}{m-1}}}
3755:
Lexicographically smallest code-word representation of a
4265:
4263:
4195:
Schumacher, Patrik (2012). "Excursion: Network Theory".
4100:
Gross, Jonathan L.; Yellen, Jay; Anderson, Mark (2018).
1522:{\textstyle \left\lfloor {\frac {i-1}{m}}\right\rfloor }
1097:{\textstyle \{1,\dots ,\lfloor {\frac {m}{2}}\rfloor \}}
4224:
Concrete
Mathematics: A Foundation for Computer Science
1798:
1727:
1492:
1266:
1196:
1148:{\textstyle \{\lceil {\frac {m}{2}}\rceil ,\dots ,m\}}
1110:
1059:
920:
846:
261:
183:, the upper bound for the maximum number of leaves is
130:-ary tree where within each level every node has 0 or
3613:
3603:
non-negative integers is a valid representation of a
3553:
3522:
3429:
3348:
3322:
3284:
3250:
3223:
3170:
3143:
3116:
3057:
2961:
2941:
2921:
2901:
2875:
2855:
2835:
2815:
2795:
2771:
2751:
2731:
2694:
2661:
2463:
2437:
2176:
2150:
2052:
2023:
1972:
1889:
1619:
1584:
1540:
1459:
678:
358:
237:
189:
4235:
4233:
3020:β 1: L-t rotation at nodes at depth
2493:
The pseudocode for this enumeration is given below:
896:{\textstyle \lfloor \log _{m}((m-1)\cdot n)\rfloor }
4755:
4634:
4581:
4510:
4367:
3164:have one child connected to their left most (i.e.,
224:The height of a tree is equal to the maximum depth
3745:
3595:
3535:
3481:
3399:
3334:
3297:
3262:
3236:
3185:
3156:
3129:
3102:
2967:
2947:
2927:
2907:
2887:
2861:
2841:
2821:
2801:
2777:
2757:
2737:
2709:
2676:
2475:
2449:
2192:
2162:
2132:
2038:
2005:
1954:
1871:
1784:
1640:
1590:
1562:
1521:
1477:
1381:
1252:
1147:
1096:
1002:
895:
821:
664:
337:
243:
202:
4298:An Introduction to Data Structures and Algorithms
1605:Pointer-based implementation of m-ary tree where
3419:rotations that has occurred at the root (i.e.,
2427:Going back to the table of enumeration of all
1485:, while its parent (if any) is found at index
1253:{\textstyle O(\log _{m}n)\equiv O(\log _{2}n)}
672:By the definition of Big-Ξ©, the maximum depth
4345:
4106:Rooted Trees, Ordered Trees, and Binary Trees
994:
973:
8:
1142:
1127:
1114:
1111:
1104:nodes would constitute the left subtree and
1091:
1088:
1075:
1060:
890:
847:
2170:nodes so that there is room for adding the
4352:
4338:
4330:
3400:{\displaystyle c_{1},c_{2},\dots ,c_{m-1}}
1155:nodes would constitute the right subtree.
4325:-ary trees, Bruno R. Preiss, Ph.D, P.Eng.
3660:
3650:
3639:
3629:
3618:
3612:
3552:
3527:
3521:
3434:
3428:
3385:
3366:
3353:
3347:
3321:
3289:
3283:
3249:
3228:
3222:
3169:
3148:
3142:
3121:
3115:
3094:
3075:
3062:
3056:
2960:
2940:
2920:
2900:
2874:
2854:
2834:
2814:
2794:
2770:
2750:
2730:
2693:
2660:
2462:
2436:
2181:
2175:
2149:
2084:
2068:
2057:
2051:
2022:
1997:
1987:
1977:
1971:
1940:
1930:
1920:
1910:
1900:
1888:
1863:
1845:
1840:
1825:
1820:
1808:
1803:
1797:
1785:{\textstyle S=s_{1},s_{2},\dots ,s_{n-1}}
1770:
1751:
1738:
1726:
1618:
1583:
1551:
1539:
1497:
1491:
1458:
1367:
1334:
1305:
1290:
1271:
1265:
1235:
1207:
1195:
1117:
1109:
1078:
1058:
993:
972:
970:
934:
925:
919:
854:
845:
799:
766:
700:
677:
602:
530:
503:
454:
419:
412:
371:
364:
359:
357:
303:
296:
287:
277:
266:
260:
236:
194:
188:
3103:{\displaystyle x_{1},x_{2},\dots ,x_{n}}
2209:
1418:An example of storing a m-ary tree with
3990:
4173:
4121:
4044:
3763:nodes is all zeros and the largest is
3043:is the inverse of this operation. The
2046:. A simple zero sequence is valid if
85:) in which each node has no more than
4001:Java: Data Structures and Programming
2975:is promoted to root, as shown below:
7:
3974:Left-child right-sibling binary tree
4069:Enumerative Combinatorics, Volume I
2849:and all of its subtrees a child of
3722:
3596:{\displaystyle (n-1)\cdot (m-1)+1}
3482:{\displaystyle c_{(i-1)(m-1)+t-1}}
2986:Convert an m-ary tree to left-tree
2655:A generation algorithm that takes
2112:
1690:is representing a 3-ary tree with
1678:using the depth-first search of a
1178:Using an array for representing a
977:
25:
4102:Graph Theory and Its Applications
3197:ary tree can be transformed to a
3137:is the root and all nodes except
1701:3-ary tree with bit sequence of
1437:, and if the tree is a complete
1041:, one must define the notion of
93:is an important case where
34:An example of a m-ary tree with
4149:(4th ed.). Section B.5.3,
3721:
3705:
2111:
2006:{\displaystyle 0^{2}4^{1}3^{2}}
1053:children of a node, the first
4272:Information Processing Letters
3777:c to zero for all i from 1 to
3688:
3676:
3673:
3661:
3584:
3572:
3566:
3554:
3462:
3450:
3447:
3435:
3201:tree using sequence of finite
3180:
3174:
2704:
2698:
2671:
2665:
2105:
2093:
2033:
2027:
1635:
1623:
1557:
1544:
1247:
1228:
1219:
1200:
952:
940:
887:
878:
866:
863:
813:
787:
778:
759:
739:
724:
712:
709:
641:
626:
614:
611:
556:
544:
481:
469:
1:
4104:(3rd ed.). Section 3.2,
2869:, additionally we assign the
906:The total number of possible
4284:10.1016/0020-0190(94)00149-9
4066:Stanley, Richard P. (2011).
3493:rotations required at depth
2935:and the right most child of
1705:and Simple Zero Sequence of
4153:: MIT Press. p. 1174.
4151:Binary and positional trees
3923:One of the applications of
2955:stays attached to it while
2829:the root node, and making
2745:being one of its nodes and
1676:bit sequence representation
1641:{\displaystyle O(m\cdot n)}
4885:
4146:Introduction to Algorithms
4108:: CRC Press. p. 132.
3789:β 1 for i from 1 to n
1659:
1478:{\displaystyle m\cdot i+c}
1026:
231:The total number of nodes
81:(or, for some authors, an
52:(for nonnegative integers
4296:Storer, James A. (2001).
4014:10.1007/978-3-642-95851-9
2207:-ary trees with 4 nodes:
1662:Enumerative Combinatorics
832:The height of a complete
4808:Left-child right-sibling
4278:(5). Elsevier: 243β247.
4180:: CS1 maint: location (
4128:: CS1 maint: location (
4051:: CS1 maint: location (
3607:ary tree if and only if
3407:represent the number of
1563:{\displaystyle O(m^{n})}
1449:-th child in range {1,β¦,
1163:-ary tree to binary tree
228:of any node in the tree.
4869:Trees (data structures)
4638:data partitioning trees
4596:C-trie (compressed ADT)
3508:-ary trees for a given
3016:child of node at depth
1431:implicit data structure
157:-ary tree in which all
4254:10.1006/jagm.1999.1073
4201:. Wiley. p. 103.
3959:
3771:β1 zero on its right.
3747:
3655:
3634:
3597:
3537:
3483:
3401:
3336:
3299:
3264:
3238:
3187:
3158:
3131:
3104:
2982:
2969:
2949:
2929:
2909:
2895:left most children of
2889:
2863:
2843:
2823:
2803:
2779:
2759:
2739:
2711:
2678:
2490:
2477:
2451:
2424:
2413:
2194:
2193:{\displaystyle n^{t}h}
2164:
2134:
2079:
2040:
2007:
1956:
1873:
1786:
1709:
1694:nodes as shown below.
1642:
1610:
1592:
1564:
1523:
1479:
1423:
1383:
1254:
1175:
1149:
1098:
1019:Traversal methods for
1004:
897:
823:
666:
339:
282:
245:
204:
179:-ary tree with height
161:are at the same depth.
145:(or, less commonly, a
38:
4300:. BirkhΓ€user Boston.
4242:Journal of Algorithms
3958:
3803:Termination Condition
3748:
3635:
3614:
3598:
3538:
3536:{\displaystyle c_{i}}
3484:
3402:
3337:
3300:
3298:{\displaystyle c_{i}}
3265:
3239:
3237:{\displaystyle x_{i}}
3188:
3159:
3157:{\displaystyle x_{n}}
3132:
3130:{\displaystyle x_{1}}
3105:
2981:
2970:
2950:
2930:
2910:
2890:
2864:
2844:
2824:
2804:
2780:
2760:
2740:
2712:
2679:
2489:
2478:
2452:
2423:
2412:
2195:
2165:
2135:
2053:
2041:
2008:
1957:
1874:
1787:
1700:
1688:x=1110000100010001000
1666:Listing all possible
1643:
1604:
1593:
1565:
1532:locality of reference
1524:
1480:
1417:
1384:
1255:
1170:
1150:
1099:
1005:
898:
824:
667:
340:
262:
246:
205:
203:{\displaystyle m^{h}}
33:
4864:Trees (graph theory)
4818:Log-structured merge
4361:Tree data structures
4226:(2nd ed.). AIP.
3767:β1 ones followed by
3611:
3551:
3520:
3427:
3346:
3320:
3282:
3248:
3221:
3168:
3141:
3114:
3055:
2959:
2939:
2919:
2899:
2873:
2853:
2833:
2813:
2793:
2769:
2749:
2729:
2710:{\displaystyle O(1)}
2692:
2677:{\displaystyle O(1)}
2659:
2651:Loopless enumeration
2461:
2435:
2174:
2148:
2050:
2039:{\displaystyle O(1)}
2021:
1970:
1887:
1796:
1725:
1719:Simple Zero Sequence
1617:
1582:
1538:
1490:
1457:
1453:} is found at index
1401:Methods for storing
1264:
1194:
1108:
1057:
918:
844:
676:
356:
259:
235:
187:
4004:. Section 8.1.2.1,
3805:Terminate when c =
3335:{\displaystyle m-1}
3263:{\displaystyle m-1}
2888:{\displaystyle m-1}
2476:{\displaystyle n=4}
2450:{\displaystyle m=3}
2163:{\displaystyle n-1}
1891:1110000100010001000
1703:1110000100010001000
345:, while the height
4783:Fractal tree index
4378:associative arrays
3960:
3785:β 1) p set to
3743:
3593:
3533:
3479:
3397:
3332:
3295:
3260:
3234:
3183:
3154:
3127:
3100:
2983:
2965:
2945:
2925:
2905:
2885:
2859:
2839:
2819:
2809:is done by making
2799:
2775:
2755:
2735:
2707:
2674:
2491:
2473:
2447:
2431:-ary trees, where
2425:
2414:
2190:
2160:
2130:
2036:
2013:, which is called
2003:
1952:
1869:
1782:
1710:
1638:
1611:
1588:
1560:
1519:
1475:
1424:
1379:
1250:
1176:
1145:
1094:
1000:
893:
819:
662:
660:
335:
241:
200:
97:= 2; similarly, a
39:
4851:
4850:
4208:978-0-470-66616-6
4115:978-1-4822-4948-4
4079:978-1-107-60262-5
4023:978-3-642-95853-3
3998:Li, Liwu (1998).
3489:is the number of
3244:until all of its
3186:{\displaystyle m}
3051:is a sequence of
2968:{\displaystyle d}
2948:{\displaystyle d}
2928:{\displaystyle a}
2908:{\displaystyle d}
2862:{\displaystyle a}
2842:{\displaystyle b}
2822:{\displaystyle d}
2802:{\displaystyle a}
2778:{\displaystyle t}
2758:{\displaystyle d}
2738:{\displaystyle a}
2403:backbone template
2399:
2398:
1591:{\displaystyle m}
1513:
1358:
1329:
1125:
1086:
992:
965:
443:
401:
333:
244:{\displaystyle N}
56:) (also known as
16:(Redirected from
4876:
4354:
4347:
4340:
4331:
4311:
4288:
4287:
4267:
4258:
4257:
4237:
4228:
4227:
4219:
4213:
4212:
4192:
4186:
4185:
4179:
4171:
4169:
4167:
4140:
4134:
4133:
4127:
4119:
4097:
4091:
4090:
4088:
4086:
4063:
4057:
4056:
4050:
4042:
4040:
4038:
3995:
3969:Branching factor
3933:English alphabet
3752:
3750:
3749:
3744:
3704:
3703:
3654:
3649:
3633:
3628:
3602:
3600:
3599:
3594:
3542:
3540:
3539:
3534:
3532:
3531:
3488:
3486:
3485:
3480:
3478:
3477:
3415:rotations, ...,
3406:
3404:
3403:
3398:
3396:
3395:
3371:
3370:
3358:
3357:
3341:
3339:
3338:
3333:
3304:
3302:
3301:
3296:
3294:
3293:
3270:sub-tree become
3269:
3267:
3266:
3261:
3243:
3241:
3240:
3235:
3233:
3232:
3203:left-t rotations
3192:
3190:
3189:
3184:
3163:
3161:
3160:
3155:
3153:
3152:
3136:
3134:
3133:
3128:
3126:
3125:
3110:nodes such that
3109:
3107:
3106:
3101:
3099:
3098:
3080:
3079:
3067:
3066:
3037:right-t rotation
2974:
2972:
2971:
2966:
2954:
2952:
2951:
2946:
2934:
2932:
2931:
2926:
2914:
2912:
2911:
2906:
2894:
2892:
2891:
2886:
2868:
2866:
2865:
2860:
2848:
2846:
2845:
2840:
2828:
2826:
2825:
2820:
2808:
2806:
2805:
2800:
2784:
2782:
2781:
2776:
2764:
2762:
2761:
2756:
2744:
2742:
2741:
2736:
2717:. For a given a
2716:
2714:
2713:
2708:
2683:
2681:
2680:
2675:
2630:β 1
2558:> 0}
2482:
2480:
2479:
2474:
2456:
2454:
2453:
2448:
2210:
2199:
2197:
2196:
2191:
2186:
2185:
2169:
2167:
2166:
2161:
2139:
2137:
2136:
2131:
2089:
2088:
2078:
2067:
2045:
2043:
2042:
2037:
2012:
2010:
2009:
2004:
2002:
2001:
1992:
1991:
1982:
1981:
1961:
1959:
1958:
1953:
1945:
1944:
1935:
1934:
1925:
1924:
1915:
1914:
1905:
1904:
1878:
1876:
1875:
1870:
1868:
1867:
1858:
1857:
1856:
1855:
1832:
1831:
1830:
1829:
1815:
1814:
1813:
1812:
1791:
1789:
1788:
1783:
1781:
1780:
1756:
1755:
1743:
1742:
1647:
1645:
1644:
1639:
1597:
1595:
1594:
1589:
1569:
1567:
1566:
1561:
1556:
1555:
1528:
1526:
1525:
1520:
1518:
1514:
1509:
1498:
1484:
1482:
1481:
1476:
1388:
1386:
1385:
1380:
1372:
1371:
1359:
1357:
1346:
1335:
1330:
1328:
1317:
1306:
1295:
1294:
1276:
1275:
1259:
1257:
1256:
1251:
1240:
1239:
1212:
1211:
1154:
1152:
1151:
1146:
1126:
1118:
1103:
1101:
1100:
1095:
1087:
1079:
1009:
1007:
1006:
1001:
999:
998:
997:
988:
976:
966:
964:
935:
930:
929:
902:
900:
899:
894:
859:
858:
828:
826:
825:
820:
803:
771:
770:
752:
748:
705:
704:
671:
669:
668:
663:
661:
654:
650:
607:
606:
585:
575:
571:
535:
534:
512:
508:
507:
465:
464:
448:
444:
442:
431:
424:
423:
413:
402:
400:
389:
382:
381:
365:
362:
344:
342:
341:
336:
334:
332:
321:
314:
313:
297:
292:
291:
281:
276:
250:
248:
247:
242:
209:
207:
206:
201:
199:
198:
21:
4884:
4883:
4879:
4878:
4877:
4875:
4874:
4873:
4854:
4853:
4852:
4847:
4751:
4630:
4577:
4506:
4502:Weight-balanced
4457:Order statistic
4371:
4363:
4358:
4318:
4308:
4295:
4292:
4291:
4269:
4268:
4261:
4239:
4238:
4231:
4221:
4220:
4216:
4209:
4194:
4193:
4189:
4172:
4165:
4163:
4161:
4142:
4141:
4137:
4120:
4116:
4099:
4098:
4094:
4084:
4082:
4080:
4065:
4064:
4060:
4043:
4036:
4034:
4024:
4006:Multi-Ary Trees
3997:
3996:
3992:
3987:
3965:
3921:
3916:
3656:
3609:
3608:
3549:
3548:
3523:
3518:
3517:
3516:. But, not all
3430:
3425:
3424:
3381:
3362:
3349:
3344:
3343:
3318:
3317:
3285:
3280:
3279:
3246:
3245:
3224:
3219:
3218:
3193:) pointer. Any
3166:
3165:
3144:
3139:
3138:
3117:
3112:
3111:
3090:
3071:
3058:
3053:
3052:
3033:
3009::
2957:
2956:
2937:
2936:
2917:
2916:
2897:
2896:
2871:
2870:
2851:
2850:
2831:
2830:
2811:
2810:
2791:
2790:
2787:left-t rotation
2767:
2766:
2747:
2746:
2727:
2726:
2690:
2689:
2657:
2656:
2653:
2648:
2635:
2606:
2591:
2570:
2563:
2556:
2531:
2522:
2512:
2505:
2459:
2458:
2433:
2432:
2177:
2172:
2171:
2146:
2145:
2080:
2048:
2047:
2019:
2018:
1993:
1983:
1973:
1968:
1967:
1936:
1926:
1916:
1906:
1896:
1885:
1884:
1859:
1841:
1836:
1821:
1816:
1804:
1799:
1794:
1793:
1766:
1747:
1734:
1723:
1722:
1682:-ary tree with
1664:
1658:
1652:Enumeration of
1615:
1614:
1580:
1579:
1576:
1547:
1536:
1535:
1499:
1493:
1488:
1487:
1455:
1454:
1412:
1407:
1363:
1347:
1336:
1318:
1307:
1286:
1267:
1262:
1261:
1231:
1203:
1192:
1191:
1165:
1106:
1105:
1055:
1054:
1031:
1025:
978:
971:
939:
921:
916:
915:
910:-ary tree with
850:
842:
841:
836:-ary tree with
762:
696:
695:
691:
674:
673:
659:
658:
598:
597:
593:
583:
582:
543:
539:
526:
510:
509:
499:
450:
446:
445:
432:
415:
414:
390:
367:
366:
354:
353:
322:
299:
298:
283:
257:
256:
233:
232:
190:
185:
184:
172:
115:
28:
23:
22:
15:
12:
11:
5:
4882:
4880:
4872:
4871:
4866:
4856:
4855:
4849:
4848:
4846:
4845:
4840:
4835:
4830:
4825:
4820:
4815:
4810:
4805:
4800:
4795:
4790:
4785:
4780:
4775:
4770:
4765:
4759:
4757:
4753:
4752:
4750:
4749:
4744:
4739:
4734:
4729:
4724:
4719:
4714:
4709:
4704:
4699:
4694:
4689:
4684:
4667:
4662:
4657:
4652:
4647:
4641:
4639:
4632:
4631:
4629:
4628:
4623:
4618:
4616:Ternary search
4613:
4608:
4603:
4598:
4593:
4587:
4585:
4579:
4578:
4576:
4575:
4570:
4565:
4560:
4555:
4550:
4545:
4540:
4532:
4527:
4522:
4516:
4514:
4508:
4507:
4505:
4504:
4499:
4494:
4489:
4484:
4479:
4474:
4464:
4459:
4454:
4449:
4444:
4439:
4429:
4424:
4419:
4414:
4409:
4404:
4399:
4394:
4389:
4383:
4381:
4365:
4364:
4359:
4357:
4356:
4349:
4342:
4334:
4328:
4327:
4317:
4316:External links
4314:
4313:
4312:
4306:
4290:
4289:
4259:
4248:(1): 100β107.
4229:
4214:
4207:
4187:
4159:
4135:
4114:
4092:
4078:
4058:
4022:
3989:
3988:
3986:
3983:
3982:
3981:
3976:
3971:
3964:
3961:
3920:
3917:
3775:Initialization
3773:
3742:
3739:
3736:
3733:
3730:
3727:
3724:
3720:
3717:
3714:
3711:
3708:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3678:
3675:
3672:
3669:
3666:
3663:
3659:
3653:
3648:
3645:
3642:
3638:
3632:
3627:
3624:
3621:
3617:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3559:
3556:
3543:sequences of
3530:
3526:
3476:
3473:
3470:
3467:
3464:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3437:
3433:
3394:
3391:
3388:
3384:
3380:
3377:
3374:
3369:
3365:
3361:
3356:
3352:
3331:
3328:
3325:
3292:
3288:
3259:
3256:
3253:
3231:
3227:
3182:
3179:
3176:
3173:
3151:
3147:
3124:
3120:
3097:
3093:
3089:
3086:
3083:
3078:
3074:
3070:
3065:
3061:
2984:
2964:
2944:
2924:
2904:
2884:
2881:
2878:
2858:
2838:
2818:
2798:
2774:
2754:
2734:
2706:
2703:
2700:
2697:
2673:
2670:
2667:
2664:
2652:
2649:
2633:
2604:
2589:
2568:
2561:
2554:
2529:
2517:
2510:
2503:
2495:
2472:
2469:
2466:
2446:
2443:
2440:
2397:
2396:
2393:
2390:
2387:
2384:
2380:
2379:
2376:
2373:
2370:
2367:
2363:
2362:
2359:
2356:
2353:
2350:
2346:
2345:
2342:
2339:
2336:
2333:
2329:
2328:
2325:
2322:
2319:
2316:
2312:
2311:
2308:
2305:
2302:
2299:
2295:
2294:
2291:
2288:
2285:
2282:
2278:
2277:
2274:
2271:
2268:
2265:
2261:
2260:
2257:
2254:
2251:
2248:
2244:
2243:
2240:
2237:
2234:
2231:
2227:
2226:
2223:
2220:
2217:
2214:
2189:
2184:
2180:
2159:
2156:
2153:
2129:
2126:
2123:
2120:
2117:
2114:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2087:
2083:
2077:
2074:
2071:
2066:
2063:
2060:
2056:
2035:
2032:
2029:
2026:
2000:
1996:
1990:
1986:
1980:
1976:
1951:
1948:
1943:
1939:
1933:
1929:
1923:
1919:
1913:
1909:
1903:
1899:
1895:
1892:
1866:
1862:
1854:
1851:
1848:
1844:
1839:
1835:
1828:
1824:
1819:
1811:
1807:
1802:
1779:
1776:
1773:
1769:
1765:
1762:
1759:
1754:
1750:
1746:
1741:
1737:
1733:
1730:
1660:Main article:
1657:
1650:
1637:
1634:
1631:
1628:
1625:
1622:
1587:
1575:
1572:
1559:
1554:
1550:
1546:
1543:
1517:
1512:
1508:
1505:
1502:
1496:
1474:
1471:
1468:
1465:
1462:
1411:
1408:
1406:
1399:
1378:
1375:
1370:
1366:
1362:
1356:
1353:
1350:
1345:
1342:
1339:
1333:
1327:
1324:
1321:
1316:
1313:
1310:
1304:
1301:
1298:
1293:
1289:
1285:
1282:
1279:
1274:
1270:
1249:
1246:
1243:
1238:
1234:
1230:
1227:
1224:
1221:
1218:
1215:
1210:
1206:
1202:
1199:
1164:
1157:
1144:
1141:
1138:
1135:
1132:
1129:
1124:
1121:
1116:
1113:
1093:
1090:
1085:
1082:
1077:
1074:
1071:
1068:
1065:
1062:
1029:tree traversal
1027:Main article:
1024:
1017:
1016:
1015:
1012:Catalan number
996:
991:
987:
984:
981:
975:
969:
963:
960:
957:
954:
951:
948:
945:
942:
938:
933:
928:
924:
904:
892:
889:
886:
883:
880:
877:
874:
871:
868:
865:
862:
857:
853:
849:
818:
815:
812:
809:
806:
802:
798:
795:
792:
789:
786:
783:
780:
777:
774:
769:
765:
761:
758:
755:
751:
747:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
711:
708:
703:
699:
694:
690:
687:
684:
681:
657:
653:
649:
646:
643:
640:
637:
634:
631:
628:
625:
622:
619:
616:
613:
610:
605:
601:
596:
592:
589:
586:
584:
581:
578:
574:
570:
567:
564:
561:
558:
555:
552:
549:
546:
542:
538:
533:
529:
525:
522:
519:
516:
513:
511:
506:
502:
498:
495:
492:
489:
486:
483:
480:
477:
474:
471:
468:
463:
460:
457:
453:
449:
447:
441:
438:
435:
430:
427:
422:
418:
411:
408:
405:
399:
396:
393:
388:
385:
380:
377:
374:
370:
363:
361:
351:
350:
331:
328:
325:
320:
317:
312:
309:
306:
302:
295:
290:
286:
280:
275:
272:
269:
265:
251:in a complete
240:
229:
222:
211:
197:
193:
171:
166:Properties of
164:
163:
162:
135:
114:
107:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4881:
4870:
4867:
4865:
4862:
4861:
4859:
4844:
4841:
4839:
4836:
4834:
4831:
4829:
4826:
4824:
4821:
4819:
4816:
4814:
4811:
4809:
4806:
4804:
4801:
4799:
4796:
4794:
4793:Hash calendar
4791:
4789:
4786:
4784:
4781:
4779:
4776:
4774:
4771:
4769:
4766:
4764:
4761:
4760:
4758:
4754:
4748:
4745:
4743:
4740:
4738:
4735:
4733:
4730:
4728:
4725:
4723:
4720:
4718:
4715:
4713:
4710:
4708:
4705:
4703:
4700:
4698:
4695:
4693:
4690:
4688:
4685:
4682:
4680:
4674:
4672:
4668:
4666:
4663:
4661:
4658:
4656:
4653:
4651:
4648:
4646:
4643:
4642:
4640:
4637:
4633:
4627:
4624:
4622:
4619:
4617:
4614:
4612:
4609:
4607:
4604:
4602:
4599:
4597:
4594:
4592:
4589:
4588:
4586:
4584:
4580:
4574:
4571:
4569:
4568:van Emde Boas
4566:
4564:
4561:
4559:
4558:Skew binomial
4556:
4554:
4551:
4549:
4546:
4544:
4541:
4539:
4537:
4533:
4531:
4528:
4526:
4523:
4521:
4518:
4517:
4515:
4513:
4509:
4503:
4500:
4498:
4495:
4493:
4490:
4488:
4485:
4483:
4480:
4478:
4475:
4473:
4469:
4465:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4445:
4443:
4440:
4438:
4437:Binary search
4434:
4430:
4428:
4425:
4423:
4420:
4418:
4415:
4413:
4410:
4408:
4405:
4403:
4400:
4398:
4395:
4393:
4390:
4388:
4385:
4384:
4382:
4379:
4375:
4370:
4366:
4362:
4355:
4350:
4348:
4343:
4341:
4336:
4335:
4332:
4326:
4324:
4320:
4319:
4315:
4309:
4307:3-7643-4253-6
4303:
4299:
4294:
4293:
4285:
4281:
4277:
4273:
4266:
4264:
4260:
4255:
4251:
4247:
4243:
4236:
4234:
4230:
4225:
4218:
4215:
4210:
4204:
4200:
4199:
4191:
4188:
4183:
4177:
4162:
4160:9780262046305
4156:
4152:
4148:
4147:
4139:
4136:
4131:
4125:
4117:
4111:
4107:
4103:
4096:
4093:
4081:
4075:
4071:
4070:
4062:
4059:
4054:
4048:
4033:
4029:
4025:
4019:
4015:
4011:
4007:
4003:
4002:
3994:
3991:
3984:
3980:
3977:
3975:
3972:
3970:
3967:
3966:
3962:
3957:
3953:
3951:
3947:
3943:
3938:
3934:
3930:
3926:
3918:
3915:
3912:
3908:
3904:
3901:
3897:
3894:
3890:
3886:
3883:
3879:
3875:
3872:
3868:
3864:
3860:
3857:
3853:
3849:
3846:
3842:
3838:
3835:
3831:
3827:
3824:
3820:
3816:
3812:
3808:
3804:
3800:
3796:
3792:
3788:
3784:
3780:
3776:
3772:
3770:
3766:
3762:
3758:
3753:
3740:
3737:
3734:
3731:
3728:
3725:
3718:
3715:
3712:
3709:
3706:
3700:
3697:
3694:
3691:
3685:
3682:
3679:
3670:
3667:
3664:
3657:
3651:
3646:
3643:
3640:
3636:
3630:
3625:
3622:
3619:
3615:
3606:
3590:
3587:
3581:
3578:
3575:
3569:
3563:
3560:
3557:
3546:
3528:
3524:
3515:
3511:
3507:
3503:
3498:
3496:
3492:
3474:
3471:
3468:
3465:
3459:
3456:
3453:
3444:
3441:
3438:
3431:
3422:
3418:
3414:
3410:
3392:
3389:
3386:
3382:
3378:
3375:
3372:
3367:
3363:
3359:
3354:
3350:
3329:
3326:
3323:
3314:
3312:
3308:
3290:
3286:
3277:
3273:
3257:
3254:
3251:
3229:
3225:
3216:
3212:
3208:
3204:
3200:
3196:
3177:
3171:
3149:
3145:
3122:
3118:
3095:
3091:
3087:
3084:
3081:
3076:
3072:
3068:
3063:
3059:
3050:
3046:
3042:
3038:
3032:
3029:
3026:
3023:
3019:
3015:
3012:
3008:
3004:
3001:
2997:
2993:
2990:
2987:
2980:
2976:
2962:
2942:
2922:
2902:
2882:
2879:
2876:
2856:
2836:
2816:
2796:
2788:
2785:-th child, a
2772:
2752:
2732:
2724:
2720:
2701:
2695:
2687:
2668:
2662:
2650:
2647:
2644:
2640:
2636:
2629:
2625:
2621:
2617:
2614:
2611:
2607:
2600:
2596:
2592:
2586:
2582:
2578:
2575:
2571:
2564:
2557:
2550:
2546:
2543:
2540:finished
2539:
2536:
2532:
2526:
2520:
2516:
2509:
2502:
2498:
2494:
2488:
2484:
2470:
2467:
2464:
2444:
2441:
2438:
2430:
2422:
2418:
2411:
2407:
2404:
2394:
2391:
2388:
2385:
2382:
2381:
2377:
2374:
2371:
2368:
2365:
2364:
2360:
2357:
2354:
2351:
2348:
2347:
2343:
2340:
2337:
2334:
2331:
2330:
2326:
2323:
2320:
2317:
2314:
2313:
2309:
2306:
2303:
2300:
2297:
2296:
2292:
2289:
2286:
2283:
2280:
2279:
2275:
2272:
2269:
2266:
2263:
2262:
2258:
2255:
2252:
2249:
2246:
2245:
2241:
2238:
2235:
2232:
2229:
2228:
2224:
2221:
2218:
2215:
2212:
2211:
2208:
2206:
2201:
2187:
2182:
2178:
2157:
2154:
2151:
2143:
2127:
2124:
2121:
2118:
2115:
2108:
2102:
2099:
2096:
2090:
2085:
2081:
2075:
2072:
2069:
2064:
2061:
2058:
2054:
2030:
2024:
2016:
2015:zero sequence
1998:
1994:
1988:
1984:
1978:
1974:
1965:
1949:
1946:
1941:
1937:
1931:
1927:
1921:
1917:
1911:
1907:
1901:
1897:
1893:
1890:
1882:
1864:
1860:
1852:
1849:
1846:
1842:
1837:
1833:
1826:
1822:
1817:
1809:
1805:
1800:
1777:
1774:
1771:
1767:
1763:
1760:
1757:
1752:
1748:
1744:
1739:
1735:
1731:
1728:
1720:
1716:
1708:
1704:
1699:
1695:
1693:
1689:
1685:
1681:
1677:
1673:
1669:
1663:
1655:
1651:
1649:
1632:
1629:
1626:
1620:
1608:
1603:
1599:
1585:
1574:Pointer-based
1573:
1571:
1552:
1548:
1541:
1533:
1529:
1515:
1510:
1506:
1503:
1500:
1494:
1472:
1469:
1466:
1463:
1460:
1452:
1448:
1444:
1440:
1436:
1432:
1428:
1421:
1416:
1409:
1404:
1400:
1398:
1396:
1390:
1376:
1373:
1368:
1364:
1360:
1354:
1351:
1348:
1343:
1340:
1337:
1331:
1325:
1322:
1319:
1314:
1311:
1308:
1302:
1299:
1296:
1291:
1287:
1283:
1280:
1277:
1272:
1268:
1244:
1241:
1236:
1232:
1225:
1222:
1216:
1213:
1208:
1204:
1197:
1189:
1185:
1181:
1174:
1169:
1162:
1158:
1156:
1139:
1136:
1133:
1130:
1122:
1119:
1083:
1080:
1072:
1069:
1066:
1063:
1052:
1048:
1044:
1040:
1036:
1033:Traversing a
1030:
1022:
1018:
1013:
989:
985:
982:
979:
967:
961:
958:
955:
949:
946:
943:
936:
931:
926:
922:
913:
909:
905:
884:
881:
875:
872:
869:
860:
855:
851:
839:
835:
831:
830:
829:
816:
810:
807:
804:
800:
796:
793:
790:
784:
781:
775:
772:
767:
763:
756:
753:
749:
745:
742:
736:
733:
730:
727:
721:
718:
715:
706:
701:
697:
692:
688:
685:
682:
679:
655:
651:
647:
644:
638:
635:
632:
629:
623:
620:
617:
608:
603:
599:
594:
590:
587:
579:
576:
572:
568:
565:
562:
559:
553:
550:
547:
540:
536:
531:
527:
523:
520:
517:
514:
504:
500:
496:
493:
490:
487:
484:
478:
475:
472:
466:
461:
458:
455:
451:
439:
436:
433:
428:
425:
420:
416:
409:
406:
403:
397:
394:
391:
386:
383:
378:
375:
372:
368:
348:
329:
326:
323:
318:
315:
310:
307:
304:
300:
293:
288:
284:
278:
273:
270:
267:
263:
255:-ary tree is
254:
238:
230:
227:
223:
220:
216:
212:
195:
191:
182:
178:
174:
173:
169:
165:
160:
156:
152:
150:
144:
142:
136:
133:
129:
125:
123:
117:
116:
112:
108:
106:
104:
101:is one where
100:
96:
92:
88:
84:
80:
76:
74:
69:
67:
62:
60:
55:
51:
49:
44:
37:
32:
19:
4678:
4670:
4535:
4468:Left-leaning
4374:dynamic sets
4369:Search trees
4322:
4297:
4275:
4271:
4245:
4241:
4223:
4217:
4197:
4190:
4164:. Retrieved
4150:
4145:
4138:
4105:
4101:
4095:
4083:. Retrieved
4068:
4061:
4035:. Retrieved
4008:: Springer.
4005:
4000:
3993:
3936:
3928:
3924:
3922:
3913:
3910:
3906:
3902:
3899:
3895:
3892:
3888:
3884:
3881:
3877:
3873:
3870:
3866:
3862:
3858:
3855:
3851:
3847:
3844:
3840:
3836:
3833:
3829:
3825:
3822:
3818:
3814:
3810:
3806:
3802:
3798:
3794:
3790:
3786:
3782:
3778:
3774:
3768:
3764:
3760:
3756:
3754:
3604:
3544:
3513:
3509:
3505:
3501:
3499:
3494:
3490:
3420:
3416:
3412:
3408:
3315:
3310:
3306:
3275:
3271:
3214:
3210:
3206:
3202:
3198:
3194:
3048:
3044:
3040:
3036:
3034:
3030:
3027:
3024:
3021:
3017:
3013:
3010:
3006:
3002:
2999:
2995:
2991:
2988:
2985:
2786:
2722:
2718:
2685:
2654:
2645:
2642:
2638:
2631:
2627:
2623:
2619:
2615:
2612:
2609:
2602:
2598:
2594:
2587:
2584:
2580:
2576:
2573:
2572:β 1
2566:
2559:
2552:
2548:
2544:
2541:
2537:
2534:
2533:= 0 for all
2527:
2524:
2518:
2514:
2507:
2500:
2496:
2492:
2428:
2426:
2415:
2402:
2400:
2204:
2202:
2141:
2014:
1963:
1880:
1718:
1714:
1711:
1706:
1702:
1691:
1687:
1683:
1679:
1675:
1671:
1667:
1665:
1653:
1612:
1606:
1577:
1486:
1450:
1446:
1442:
1438:
1426:
1425:
1419:
1402:
1394:
1391:
1187:
1183:
1179:
1177:
1172:
1160:
1050:
1046:
1042:
1038:
1034:
1032:
1020:
1010:(which is a
911:
907:
837:
833:
352:
346:
252:
225:
218:
214:
180:
176:
167:
154:
153:) is a full
148:
146:
140:
138:
131:
127:
121:
119:
110:
102:
99:ternary tree
94:
89:children. A
86:
83:ordered tree
79:arborescence
77:tree) is an
72:
71:
65:
64:
58:
57:
53:
47:
46:
43:graph theory
40:
35:
4768:Exponential
4756:Other trees
3979:Binary tree
3919:Application
3411:rotations,
3278:denoted by
2998::
2601:β 1) β sum(
1422:in an array
1397:-ary tree.
213:The height
91:binary tree
4858:Categories
4712:Priority R
4462:Palindrome
3985:References
3854:] + 1
3423:=1).Then,
3305:defines a
3199:left-chain
3045:left chain
2608:)
1598:children:
1405:-ary trees
1159:Convert a
170:-ary trees
159:leaf nodes
113:-ary trees
18:K-ary tree
4798:iDistance
4677:implicit
4665:Hilbert R
4660:Cartesian
4543:Fibonacci
4477:Scapegoat
4472:Redβblack
4176:cite book
4124:cite book
4047:cite book
3813:NEXT
3811:Procedure
3735:…
3729:∈
3723:∀
3713:−
3707:≤
3698:−
3683:−
3668:−
3637:∑
3616:∑
3579:−
3570:⋅
3561:−
3472:−
3457:−
3442:−
3390:−
3376:…
3342:tuple of
3327:−
3255:−
3085:…
3025:end while
2880:−
2721:ary tree
2497:Procedure
2155:−
2125:−
2119:≤
2113:∀
2100:−
2091:≤
2055:∑
1947:≡
1894:≡
1850:−
1834:…
1775:−
1761:…
1656:ary trees
1630:⋅
1504:−
1464:⋅
1374:
1352:
1341:
1332:⋅
1323:
1312:
1297:
1284:⋅
1278:
1242:
1223:≡
1214:
1134:…
1128:⌉
1115:⌈
1089:⌋
1076:⌊
1070:…
1023:ary trees
983:⋅
968:⋅
947:−
914:nodes is
891:⌋
882:⋅
873:−
861:
848:⌊
840:nodes is
808:
794:
773:
743:−
728:⋅
719:−
707:
689:≥
645:−
630:⋅
621:−
609:
591:≥
560:⋅
551:−
537:
524:≥
485:⋅
476:−
467:≥
437:−
426:−
404:≥
395:−
384:−
327:−
316:−
264:∑
151:-ary tree
143:-ary tree
139:complete
134:children.
124:-ary tree
109:Types of
50:-ary tree
4813:Link/cut
4525:Binomial
4452:Interval
3963:See also
3909:β 1
3891:β 1
3869:β 0
3832:+ 1
3793:β 0
3316:Let the
3307:codeword
2641:β 1
2626:+ 3, β¦,
2597:+ 1) β
(
1516:⌋
1495:⌊
1039:m > 2
750:⌉
693:⌈
652:⌉
595:⌈
147:perfect
4773:Fenwick
4737:Segment
4636:Spatial
4553:Pairing
4548:Leftist
4470:)
4442:Dancing
4435:)
4433:Optimal
4166:20 July
4085:20 July
4037:20 July
3948:and/or
3839:] >
3031:end for
3028:end for
2547:β max {
175:For an
4823:Merkle
4788:Fusion
4778:Finger
4702:Octree
4692:Metric
4626:Y-fast
4621:X-fast
4611:Suffix
4530:Brodal
4520:Binary
4304:
4205:
4157:
4112:
4076:
4032:708416
4030:
4020:
3946:Octree
3942:B-tree
3911:end if
3865:]
3856:end if
3843:] + 1
3821:+ 1 β
3005:= 2...
2994:= 1...
2643:end if
2610:end if
2523:)
1879:where
1707:004433
1445:, its
1435:arrays
1410:Arrays
1260:since
217:of an
126:is an
4833:Range
4803:K-ary
4763:Cover
4606:Radix
4591:Ctrie
4583:Tries
4512:Heaps
4492:Treap
4482:Splay
4447:HTree
4402:(a,b)
4392:2β3β4
4028:S2CID
3861:]] β
3759:with
3757:m-ary
3309:of a
3209:from
3011:while
2725:with
2622:+ 2,
2579:<
2513:, β¦,
2499:NEXT(
1964:00433
1950:00433
1047:right
120:full
105:= 3.
45:, an
4838:SPQR
4717:Quad
4645:Ball
4601:Hash
4573:Weak
4563:Skew
4538:-ary
4302:ISBN
4203:ISBN
4182:link
4168:2023
4155:ISBN
4130:link
4110:ISBN
4087:2023
4074:ISBN
4053:link
4039:2023
4018:ISBN
3950:trie
3893:else
3882:then
3850:] β
3845:then
3809:β 1
3801:β 1
3512:and
3272:null
3205:for
2765:its
2585:then
2583:β 1
2542:else
2538:then
2457:and
2395:000
2378:001
2361:002
2344:003
2327:004
2310:005
2293:006
2276:010
2259:011
2242:012
2225:013
1045:and
1043:left
577:>
497:>
410:>
75:-way
68:-ary
61:-ary
4843:Top
4697:MVP
4655:BSP
4407:AVL
4387:2β3
4280:doi
4250:doi
4010:doi
3914:end
3900:sum
3874:sum
3819:sum
3815:sum
3791:sum
3491:L-t
3417:L-m
3413:L-3
3409:L-2
3213:to
3047:of
3039:at
3000:for
2989:for
2915:to
2789:at
2646:end
2613:for
2593:β (
2392:014
2389:040
2386:112
2383:201
2375:015
2372:041
2369:113
2366:202
2358:020
2355:042
2352:114
2349:203
2341:021
2338:100
2335:120
2332:204
2324:022
2321:101
2318:121
2315:210
2307:023
2304:102
2301:122
2298:211
2290:024
2287:103
2284:123
2281:212
2273:030
2270:104
2267:130
2264:213
2256:031
2253:105
2250:131
2247:220
2239:032
2236:110
2233:132
2230:221
2222:033
2219:111
2216:200
2213:222
1966:is
1692:n=6
1609:=4.
1433:in
1420:m=3
1365:log
1349:log
1338:log
1320:log
1309:log
1288:log
1269:log
1233:log
1205:log
1173:m=6
852:log
805:log
791:log
764:log
698:log
600:log
528:log
70:or
41:In
36:m=5
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