484:
472:
1097:
1118:
may be absorbed by reorganizing the tree, or it may repeatedly travel upwards before it can be absorbed, as a temporary 4-node may in the case of insertion. Alternatively, it's possible to use an algorithm which is both top-down and bottom-up, creating temporary 4-nodes on the way down that are then destroyed as you travel back up. Deletion methods are explained in more detail in the references.
1108:
If the target node is a 3-node and the parent is a 3-node, a temporary 4-node is created then split as above. This process continues up the tree to the root. If the root must be split, then the process of a single 3-node is followed: a temporary 4-node root is split into three 2-nodes, one of which
1117:
Deleting a key from a non-leaf node can be done by replacing it by its immediate predecessor or successor, and then deleting the predecessor or successor from a leaf node. Deleting a key from a leaf node is easy if the leaf is a 3-node. Otherwise, it may require creating a temporary 1-node which
1104:
If the target node is a 3-node whose parent is a 2-node, the key is inserted into the 3-node to create a temporary 4-node. In the illustration, the key 10 is inserted into the 2-node with 6 and 9. The middle key is 9, and is promoted to the parent 2-node. This leaves a 3-node of 6 and 10, which is
1092:
To insert into a 3-node, more work may be required depending on the location of the 3-node. If the tree consists only of a 3-node, the node is split into three 2-nodes with the appropriate keys and children.
422:
2–3 trees are required to be balanced, meaning that each leaf is at the same level. It follows that each right, center, and left subtree of a node contains the same or close to the same amount of data.
1028:
689:. Since the data elements in each node are ordered, a search function will be directed to the correct subtree and eventually to the correct node which contains the item.
372:
336:
293:
257:
214:
178:
1065:
985:
933:
866:
829:
105:
2214:
1890:
465:, with three data elements, may be temporarily created during manipulation of the tree but is never persistently stored in the tree.
2184:
1615:
1326:, Lyn Turbak, handout #26, course notes, CS230 Data Structures, Wellesley College, December 2, 2004. Accessed Mar. 11, 2024.
1308:
1278:
1197:
1822:
2253:
1366:
2123:
1225:
1913:
1482:
1918:
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1992:
400:
1131:
132:
1997:
1980:
546:
1323:
2196:
1963:
1958:
1447:
1953:
1832:
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1270:
2227:
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1001:
396:
27:
2209:
2009:
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227:
184:
148:
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1526:
1375:
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845:
808:
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1471:
81:
8:
1630:
1263:
1243:
1204:
The 2–3 trees defined at the close of
Section 6.2.3 are equivalent to B-Trees of order 3.
392:
2118:
2103:
1968:
1928:
1797:
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1582:
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1451:
1406:
1143:
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686:
125:
1089:
To insert into a 2-node, the new key is added to the 2-node in the appropriate order.
2037:
1936:
1691:
1392:
1352:
1304:
1274:
1221:
1193:
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471:
384:
67:
1109:
is considered to be the root. This operation grows the height of the tree by one.
2248:
2080:
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113:
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2150:
1975:
1899:
1679:
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56:
2242:
2145:
2042:
2027:
1827:
1807:
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1466:
685:
Searching for an item in a 2–3 tree is similar to searching for an item in a
416:
48:
415:) have no children and one or two data elements. 2–3 trees were invented by
1787:
1751:
1567:
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1544:
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404:
2140:
2065:
1837:
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2017:
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1741:
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1868:
407:
or three children (3-node) and two data elements. A 2–3 tree is a
2162:
2108:
1842:
1746:
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1411:
1153:
1096:
2157:
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1426:
1158:
1105:
split to be two 2-nodes held as children of the parent 3-node.
408:
1241:
Aho, Alfred V.; Hopcroft, John E.; Ullman, Jeffrey D. (1974).
1605:
1506:
1461:
2179:
1597:
839:, which by definition is a 2–3 tree, and go back to step 2.
1100:
Insertion of a number in a 2–3 tree for 3 possible cases
1216:
R. Hernández; J. C. Lázaro; R. Dormido; S. Ros (2001).
1086:
Insertion maintains the balanced property of the tree.
1047:
1004:
967:
915:
848:
811:
502:
if and only if one of the following statements hold:
345:
309:
266:
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187:
151:
84:
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1059:
1022:
979:
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823:
366:
330:
287:
251:
208:
172:
99:
1240:
2240:
1245:The Design and Analysis of Computer Algorithms
411:of order 3. Nodes on the outside of the tree (
1884:
1360:
1192:. Vol. 3 (2 ed.). Addison Wesley.
1126:Since 2–3 trees are similar in structure to
665:Every internal node is a 2-node or a 3-node.
403:) has either two children (2-node) and one
1891:
1877:
1367:
1353:
1298:
758:. We need no further steps and we're done.
1374:
1299:Sedgewick, Robert; Wayne, Kevin. "3.3".
1095:
701:be the data element we want to find. If
1132:parallel algorithms for red–black trees
2241:
1260:
1121:
1872:
1348:
1187:
1134:can be applied to 2–3 trees as well.
645:is greater than each data element in
631:is greater than each data element in
1294:
1292:
1290:
1898:
649:and less than each data element in
635:and less than each data element in
13:
1269:. London: The MIT Press. pp.
565:is less than each data element in
446:We say that an internal node is a
431:We say that an internal node is a
16:Data structure in computer science
14:
2265:
1333:
1287:
1188:Knuth, Donald M (1998). "6.2.4".
671:All data is kept in sorted order.
668:All leaves are at the same level.
1218:Estructura de Datos y Algoritmos
555:is greater than each element in
482:
470:
1190:The Art of Computer Programming
577:is a 3-node with data elements
1317:
1303:(4 ed.). Addison Wesley.
1254:
1234:
1209:
1181:
625:are 2–3 trees of equal height;
519:is a 2-node with data element
426:
361:
349:
325:
313:
282:
270:
246:
234:
203:
191:
167:
155:
94:
88:
1:
1340:2–3 Tree In-depth description
1174:
675:
659:
1081:
905:be the two data elements of
885:is a 3-node with left child
767:is a 2-node with left child
680:
7:
2215:Directed acyclic word graph
1981:Double-ended priority queue
1137:
1112:
1023:{\displaystyle a<d<b}
10:
2270:
1265:Introduction to Algorithms
545:are 2–3 trees of the same
509:is empty. In other words,
2254:Amortized data structures
2223:
2195:
2089:
2051:
2008:
1927:
1906:
1770:
1649:
1596:
1525:
1382:
783:. There are three cases:
367:{\displaystyle O(\log n)}
331:{\displaystyle O(\log n)}
298:
288:{\displaystyle O(\log n)}
252:{\displaystyle O(\log n)}
219:
209:{\displaystyle O(\log n)}
173:{\displaystyle O(\log n)}
140:
119:
112:
73:
66:
62:
54:
44:
36:
26:
21:
1947:Retrieval Data Structure
1823:Left-child right-sibling
935:. There are four cases:
513:does not have any nodes.
2228:List of data structures
2205:Binary decision diagram
1653:data partitioning trees
1611:C-trie (compressed ADT)
1261:Cormen, Thomas (2009).
779:be the data element in
2210:Directed acyclic graph
1101:
1075:and go back to step 2.
1061:
1060:{\displaystyle d>b}
1038:and go back to step 2.
1024:
995:and go back to step 2.
981:
980:{\displaystyle d<a}
929:
928:{\displaystyle a<b}
876:and go back to step 2.
862:
861:{\displaystyle d>a}
825:
824:{\displaystyle d<a}
368:
332:
289:
253:
210:
174:
101:
1099:
1062:
1025:
982:
930:
863:
826:
369:
333:
290:
254:
211:
175:
102:
2076:Unrolled linked list
1833:Log-structured merge
1376:Tree data structures
1045:
1002:
965:
913:
846:
809:
343:
307:
264:
228:
185:
149:
100:{\displaystyle O(n)}
82:
2124:Self-balancing tree
1122:Parallel operations
794:, then we've found
393:tree data structure
2104:Binary search tree
1969:Double-ended queue
1798:Fractal tree index
1393:associative arrays
1102:
1057:
1020:
977:
925:
893:, and right child
858:
821:
697:be a 2–3 tree and
687:binary search tree
607:, and right child
454:data elements and
364:
328:
285:
249:
206:
170:
97:
2236:
2235:
2038:Hashed array tree
1937:Associative array
1866:
1865:
1310:978-0-321-57351-3
1280:978-0-262-03384-8
1249:. Addison-Wesley.
1220:. Prentice Hall.
1199:978-0-201-89685-5
439:data element and
381:
380:
377:
376:
2261:
2061:Association list
1893:
1886:
1879:
1870:
1869:
1369:
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1315:
1314:
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949:
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871:
867:
865:
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859:
838:
834:
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828:
827:
822:
801:
797:
793:
789:
782:
778:
774:
771:and right child
770:
766:
757:
753:
749:
745:
741:
737:
730:
723:
719:
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584:
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558:
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531:and right child
530:
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512:
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497:
486:
474:
385:computer science
373:
371:
370:
365:
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255:
250:
215:
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207:
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106:
104:
103:
98:
68:Space complexity
64:
63:
55:Complexities in
19:
18:
2269:
2268:
2264:
2263:
2262:
2260:
2259:
2258:
2239:
2238:
2237:
2232:
2219:
2191:
2085:
2081:XOR linked list
2047:
2023:Circular buffer
2004:
1923:
1902:
1900:Data structures
1897:
1867:
1862:
1766:
1645:
1592:
1521:
1517:Weight-balanced
1472:Order statistic
1386:
1378:
1373:
1336:
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1311:
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1214:
1210:
1200:
1186:
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1140:
1128:red–black trees
1124:
1115:
1084:
1072:
1068:
1046:
1043:
1042:
1035:
1031:
1003:
1000:
999:
992:
988:
966:
963:
962:
958:and we're done.
955:
951:
947:
943:
939:
914:
911:
910:
906:
902:
898:
894:
890:
889:, middle child
886:
882:
873:
869:
847:
844:
843:
836:
832:
810:
807:
806:
802:and we're done.
799:
795:
791:
787:
780:
776:
772:
768:
764:
755:
751:
747:
743:
739:
735:
728:
721:
720:be the root of
717:
713:and we're done.
710:
706:
705:is empty, then
702:
698:
694:
683:
678:
662:
650:
646:
642:
636:
632:
628:
622:
618:
614:
608:
604:
603:, middle child
600:
599:has left child
596:
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582:
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566:
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556:
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527:has left child
524:
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429:
399:with children (
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114:Time complexity
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2173:Hilbert R-tree
2170:
2165:
2155:
2154:
2153:
2151:Fibonacci heap
2148:
2143:
2133:
2132:
2131:
2126:
2121:
2119:Red–black tree
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2111:
2101:
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1976:Priority queue
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1631:Ternary search
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1334:External links
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57:big O notation
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2149:
2147:
2146:Binomial heap
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2043:Sparse matrix
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2029:
2028:Dynamic array
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1808:Hash calendar
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1601:
1599:
1595:
1589:
1586:
1584:
1583:van Emde Boas
1581:
1579:
1576:
1574:
1573:Skew binomial
1571:
1569:
1566:
1564:
1561:
1559:
1556:
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1478:
1475:
1473:
1470:
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1463:
1460:
1458:
1455:
1453:
1452:Binary search
1449:
1445:
1443:
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1438:
1435:
1433:
1430:
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1425:
1423:
1420:
1418:
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1237:
1229:
1227:84-205-2980-X
1223:
1219:
1212:
1205:
1201:
1195:
1191:
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997:
974:
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922:
919:
916:
880:
855:
852:
849:
841:
818:
815:
812:
804:
785:
784:
762:
750:. Otherwise,
733:
732:
726:
715:
692:
691:
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688:
670:
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664:
663:
641:
627:
613:
612:
573:
561:
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468:
467:
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457:
453:
449:
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442:
438:
434:
424:
420:
418:
417:John Hopcroft
414:
410:
406:
402:
401:internal node
398:
394:
390:
386:
358:
355:
352:
346:
339:
322:
319:
316:
310:
303:
301:
297:
279:
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267:
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243:
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224:
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200:
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145:
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118:
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111:
108:
91:
85:
78:
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72:
69:
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61:
58:
53:
50:
49:John Hopcroft
47:
43:
39:
35:
31:
29:
25:
20:
1998:Disjoint-set
1693:
1685:
1550:
1483:Left-leaning
1401:
1389:dynamic sets
1384:Search trees
1319:
1300:
1264:
1256:
1251:, pp.145–147
1244:
1236:
1217:
1211:
1203:
1189:
1183:
1125:
1116:
1107:
1103:
1091:
1088:
1085:
942:is equal to
790:is equal to
684:
499:
494:We say that
493:
462:
460:
455:
451:
447:
445:
440:
436:
432:
430:
421:
405:data element
388:
382:
133:
126:
2141:Binary heap
2066:Linked list
1783:Exponential
1771:Other trees
1324:"2-3 Trees"
1169:Finger tree
1067:, then set
1030:, then set
987:, then set
868:, then set
831:, then set
731:is a leaf.
427:Definitions
45:Invented by
2243:Categories
2129:Splay tree
2033:Hash table
1914:Collection
1727:Priority R
1477:Palindrome
1301:Algorithms
1175:References
1164:(a,b)-tree
1144:2–3–4 tree
746:is not in
738:is not in
709:is not in
676:Operations
660:Properties
458:children.
450:if it has
443:children.
435:if it has
413:leaf nodes
134:Worst Case
2185:Hash tree
2071:Skip list
2018:Bit array
1919:Container
1813:iDistance
1692:implicit
1680:Hilbert R
1675:Cartesian
1558:Fibonacci
1492:Scapegoat
1487:Red–black
1082:Insertion
681:Searching
419:in 1970.
356:
320:
277:
241:
198:
162:
127:Amortized
2114:AVL tree
1993:Multiset
1942:Multimap
1929:Abstract
1828:Link/cut
1540:Binomial
1467:Interval
1149:2–3 heap
1138:See also
1113:Deletion
909:, where
881:Suppose
763:Suppose
727:Suppose
585:, where
500:2–3 tree
389:2–3 tree
120:Function
37:Invented
22:2–3 tree
2168:R+ tree
2163:R* tree
2109:AA tree
1788:Fenwick
1752:Segment
1651:Spatial
1568:Pairing
1563:Leftist
1485:)
1457:Dancing
1450:)
1448:Optimal
1154:AA tree
950:, then
742:, then
611:, then
535:, then
2249:B-tree
2197:Graphs
2158:R-tree
2099:B-tree
2053:Linked
2010:Arrays
1838:Merkle
1803:Fusion
1793:Finger
1717:Octree
1707:Metric
1641:Y-fast
1636:X-fast
1626:Suffix
1545:Brodal
1535:Binary
1307:
1277:
1224:
1196:
1159:B-tree
954:is in
897:. Let
775:. Let
754:is in
621:, and
547:height
489:3 node
477:2 node
463:4-node
448:3-node
433:2-node
409:B-tree
299:Delete
220:Insert
141:Search
2091:Trees
1964:Queue
1959:Stack
1907:Types
1848:Range
1818:K-ary
1778:Cover
1621:Radix
1606:Ctrie
1598:Tries
1527:Heaps
1507:Treap
1497:Splay
1462:HTree
1417:(a,b)
1407:2–3–4
639:; and
595:. If
559:; and
523:. If
498:is a
456:three
391:is a
74:Space
2180:Trie
2136:Heap
1954:List
1853:SPQR
1732:Quad
1660:Ball
1616:Hash
1588:Weak
1578:Skew
1553:-ary
1305:ISBN
1275:ISBN
1222:ISBN
1194:ISBN
1052:>
1015:<
1009:<
972:<
920:<
901:and
853:>
816:<
716:Let
693:Let
590:<
581:and
541:and
397:node
387:, a
40:1970
32:tree
28:Type
1988:Set
1858:Top
1712:MVP
1670:BSP
1422:AVL
1402:2–3
1271:504
1071:to
1041:If
1034:to
998:If
991:to
961:If
946:or
938:If
872:to
842:If
835:to
805:If
798:in
786:If
734:If
452:two
441:two
437:one
383:In
353:log
317:log
274:log
238:log
195:log
159:log
2245::
1843:PQ
1757:VP
1747:R*
1742:R+
1722:PH
1696:-d
1688:-d
1665:BK
1512:UB
1437:B*
1432:B+
1412:AA
1289:^
1273:.
1202:.
1130:,
617:,
461:A
1892:e
1885:t
1878:v
1762:X
1737:R
1702:M
1698:)
1694:k
1690:(
1686:k
1551:d
1502:T
1481:(
1446:(
1442:B
1427:B
1395:)
1391:/
1387:(
1368:e
1361:t
1354:v
1313:.
1283:.
1230:.
1073:r
1069:T
1055:b
1049:d
1036:q
1032:T
1018:b
1012:d
1006:a
993:p
989:T
975:a
969:d
956:T
952:d
948:b
944:a
940:d
923:b
917:a
907:t
903:b
899:a
895:r
891:q
887:p
883:t
874:q
870:T
856:a
850:d
837:p
833:T
819:a
813:d
800:T
796:d
792:a
788:d
781:t
777:a
773:q
769:p
765:t
756:T
752:d
748:T
744:d
740:t
736:d
729:t
724:.
722:T
718:t
711:T
707:d
703:T
699:d
695:T
653:.
651:r
647:q
643:b
637:q
633:p
629:a
623:r
619:q
615:p
609:r
605:q
601:p
597:T
593:b
587:a
583:b
579:a
575:T
569:.
567:q
563:a
557:p
553:a
549:;
543:q
539:p
533:q
529:p
525:T
521:a
517:T
511:T
507:T
496:T
362:)
359:n
350:(
347:O
326:)
323:n
314:(
311:O
283:)
280:n
271:(
268:O
247:)
244:n
235:(
232:O
204:)
201:n
192:(
189:O
168:)
165:n
156:(
153:O
95:)
92:n
89:(
86:O
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