1066:. The empty sequence is the unique minimal element, and each element has a finite and well-ordered set of predecessors (the set of all of its prefixes). An order-theoretic tree may be represented by an isomorphic tree of sequences if and only if each of its elements has finite height (that is, a finite set of predecessors).
1867:
937:, and an outgoing edge from each nonempty sequence that connects it to the shorter sequence formed by removing its last element. This graph is a tree in the graph-theoretic sense. The root of the tree is the empty sequence.
1557:
871:
748:
404:
261:
1603:
897:
in which every vertex except for a special root vertex has exactly one outgoing edge, and in which the path formed by following these edges from any vertex eventually leads to the root vertex. If
1725:
1404:
190:
140:
333:
1178:
1119:
1342:
1661:
1024:
450:
1444:
1424:
1362:
1310:
1264:
1244:
1224:
1151:
1092:
1064:
1044:
998:
978:
935:
915:
788:
768:
671:
647:
594:
542:
522:
502:
482:
424:
301:
281:
160:
110:
73:
49:
1717:
1687:
1290:
1204:
568:
1449:
1605:(the subset for which the length of the first sequence is equal to or 1 more than the length of the second sequence). In this way we may identify
960:
set of predecessors. Every tree in descriptive set theory is also an order-theoretic tree, using a partial ordering in which two sequences
793:
16:
This article is about mathematical trees described by prefixes of finite sequences. For trees described by partially ordered sets, see
676:
338:
195:
1862:{\displaystyle p=\{{\vec {x}}\in X^{\omega }|(\exists {\vec {y}}\in Y^{\omega })\langle {\vec {x}},{\vec {y}}\rangle \in \}}
1562:
1559:). Elements in this subspace are identified in the natural way with a subset of the product of two spaces of sequences,
1923:
1916:
917:
is a tree in the descriptive set theory sense, then it corresponds to a graph with one vertex for each sequence in
1940:
1367:
1908:
1945:
165:
115:
76:
306:
1156:
1097:
1321:
24:
1887:
1608:
949:
1003:
1950:
1900:
8:
429:
614:
1429:
1409:
1347:
1295:
1249:
1229:
1209:
1136:
1077:
1049:
1029:
983:
963:
920:
900:
773:
753:
656:
632:
579:
527:
507:
487:
467:
409:
286:
266:
145:
95:
58:
34:
1696:
1666:
1269:
1183:
547:
1919:
1912:
1316:
945:
17:
1122:
953:
52:
1130:
894:
1552:{\displaystyle \langle x_{0},y_{1},x_{2},y_{3}\ldots ,x_{2m},y_{2m+1}\rangle }
1934:
957:
941:
886:
1344:
are considered. In this case, by convention, we consider only the subset
890:
602:
1883:
1879:
618:
621:
with an infinite number of sequences must necessarily be illfounded.
1246:
consist of the set of finite prefixes of the infinite sequences in
866:{\displaystyle \langle x_{0},x_{1},\ldots ,x_{n-1},x\rangle \in T}
743:{\displaystyle \langle x_{0},x_{1},\ldots ,x_{n-1}\rangle \in T}
79:
of a sequence in the collection also belongs to the collection.
92:
The collection of all finite sequences of elements of a set
399:{\displaystyle \langle x_{0},x_{1},\ldots ,x_{m-1}\rangle }
256:{\displaystyle \langle x_{0},x_{1},\ldots ,x_{n-1}\rangle }
1406:, containing only sequences whose even elements come from
873:. A tree that does not have any terminal nodes is called
1728:
1699:
1669:
1611:
1598:{\displaystyle X^{<\omega }\times Y^{<\omega }}
1565:
1452:
1432:
1412:
1370:
1350:
1324:
1298:
1272:
1252:
1232:
1212:
1186:
1159:
1139:
1100:
1080:
1052:
1032:
1006:
986:
966:
923:
903:
796:
776:
756:
679:
659:
635:
582:
550:
530:
510:
490:
470:
452:
shows that the empty sequence belongs to every tree.
432:
412:
341:
309:
289:
269:
198:
168:
148:
118:
98:
61:
37:
1689:for over the product space. We may then form the
880:
1861:
1711:
1681:
1655:
1597:
1551:
1438:
1418:
1398:
1356:
1336:
1304:
1284:
1258:
1238:
1218:
1198:
1172:
1145:
1113:
1086:
1058:
1038:
1018:
992:
972:
929:
909:
865:
782:
762:
742:
665:
641:
588:
562:
536:
516:
496:
476:
444:
418:
398:
327:
295:
275:
255:
184:
154:
142:. With this notation, a tree is a nonempty subset
134:
104:
67:
43:
1932:
653:if it is not a prefix of a longer sequence in
1856:
1841:
1811:
1744:
1546:
1453:
854:
797:
731:
680:
393:
342:
250:
199:
944:, a different notion of a tree is used: an
504:, each of whose finite prefixes belongs to
1399:{\displaystyle (X\times Y)^{<\omega }}
629:A finite sequence that belongs to a tree
1133:. In this topology, every closed subset
1899:
484:is an infinite sequence of elements of
1933:
599:A tree that has no branches is called
455:
1312:forms a closed set in this topology.
607:; a tree with at least one branch is
1074:The set of infinite sequences over
750:is terminal if there is no element
13:
1780:
524:. The set of all branches through
14:
1962:
624:
1905:Classical Descriptive Set Theory
881:Relation to other types of trees
185:{\displaystyle X^{<\omega }}
135:{\displaystyle X^{<\omega }}
1853:
1847:
1835:
1820:
1808:
1789:
1777:
1773:
1753:
1738:
1732:
1706:
1700:
1676:
1670:
1650:
1634:
1628:
1612:
1384:
1371:
1279:
1273:
1193:
1187:
557:
551:
335:, then the shortened sequence
82:
1:
1909:Graduate Texts in Mathematics
1893:
956:in which each element has a
328:{\displaystyle 0\leq m<n}
7:
1873:
1426:and odd elements come from
1173:{\displaystyle X^{\omega }}
1114:{\displaystyle X^{\omega }}
1069:
10:
1967:
426:. In particular, choosing
15:
1882:, a type of tree used in
1337:{\displaystyle X\times Y}
263:is a sequence of length
87:
1886:as part of a notion of
1656:{\displaystyle \times }
1266:. Conversely, the body
1941:Descriptive set theory
1863:
1713:
1683:
1657:
1599:
1553:
1440:
1420:
1400:
1364:of the product space,
1358:
1338:
1306:
1286:
1260:
1240:
1220:
1200:
1174:
1147:
1115:
1088:
1060:
1046:is a proper prefix of
1040:
1020:
1019:{\displaystyle T<U}
994:
974:
931:
911:
867:
784:
764:
744:
667:
643:
590:
564:
538:
518:
498:
478:
446:
420:
400:
329:
297:
277:
257:
186:
156:
136:
106:
69:
45:
25:descriptive set theory
1901:Kechris, Alexander S.
1864:
1714:
1684:
1658:
1600:
1554:
1441:
1421:
1401:
1359:
1339:
1307:
1287:
1261:
1241:
1221:
1206:for some pruned tree
1201:
1175:
1148:
1116:
1089:
1061:
1041:
1021:
995:
975:
950:partially ordered set
932:
912:
868:
785:
765:
745:
668:
644:
591:
565:
539:
519:
499:
479:
447:
421:
401:
330:
298:
278:
258:
187:
157:
137:
107:
70:
46:
1726:
1697:
1667:
1609:
1563:
1450:
1430:
1410:
1368:
1348:
1322:
1315:Frequently trees on
1296:
1270:
1250:
1230:
1210:
1184:
1157:
1137:
1098:
1078:
1050:
1030:
1004:
984:
964:
946:order-theoretic tree
921:
901:
794:
774:
754:
677:
657:
633:
580:
548:
528:
508:
488:
468:
430:
410:
339:
307:
287:
267:
196:
166:
146:
116:
96:
59:
35:
1121:) may be given the
456:Branches and bodies
445:{\displaystyle m=0}
51:is a collection of
1946:Trees (set theory)
1859:
1709:
1679:
1653:
1595:
1549:
1436:
1416:
1396:
1354:
1334:
1317:Cartesian products
1302:
1282:
1256:
1236:
1216:
1196:
1170:
1143:
1111:
1084:
1056:
1036:
1016:
990:
970:
927:
907:
863:
780:
760:
740:
663:
639:
586:
560:
534:
514:
494:
474:
442:
416:
396:
325:
293:
273:
253:
182:
152:
132:
102:
65:
41:
1838:
1823:
1792:
1756:
1439:{\displaystyle Y}
1419:{\displaystyle X}
1357:{\displaystyle T}
1305:{\displaystyle T}
1259:{\displaystyle C}
1239:{\displaystyle T}
1219:{\displaystyle T}
1146:{\displaystyle C}
1087:{\displaystyle X}
1059:{\displaystyle U}
1039:{\displaystyle T}
993:{\displaystyle U}
973:{\displaystyle T}
930:{\displaystyle T}
910:{\displaystyle T}
783:{\displaystyle X}
763:{\displaystyle x}
666:{\displaystyle T}
642:{\displaystyle T}
589:{\displaystyle T}
537:{\displaystyle T}
517:{\displaystyle T}
497:{\displaystyle X}
477:{\displaystyle T}
419:{\displaystyle T}
296:{\displaystyle T}
276:{\displaystyle n}
155:{\displaystyle T}
105:{\displaystyle X}
68:{\displaystyle X}
44:{\displaystyle X}
18:Tree (set theory)
1958:
1927:
1868:
1866:
1865:
1860:
1840:
1839:
1831:
1825:
1824:
1816:
1807:
1806:
1794:
1793:
1785:
1776:
1771:
1770:
1758:
1757:
1749:
1718:
1716:
1715:
1712:{\displaystyle }
1710:
1688:
1686:
1685:
1682:{\displaystyle }
1680:
1662:
1660:
1659:
1654:
1649:
1648:
1627:
1626:
1604:
1602:
1601:
1596:
1594:
1593:
1578:
1577:
1558:
1556:
1555:
1550:
1545:
1544:
1523:
1522:
1504:
1503:
1491:
1490:
1478:
1477:
1465:
1464:
1445:
1443:
1442:
1437:
1425:
1423:
1422:
1417:
1405:
1403:
1402:
1397:
1395:
1394:
1363:
1361:
1360:
1355:
1343:
1341:
1340:
1335:
1311:
1309:
1308:
1303:
1291:
1289:
1288:
1285:{\displaystyle }
1283:
1265:
1263:
1262:
1257:
1245:
1243:
1242:
1237:
1225:
1223:
1222:
1217:
1205:
1203:
1202:
1199:{\displaystyle }
1197:
1179:
1177:
1176:
1171:
1169:
1168:
1152:
1150:
1149:
1144:
1123:product topology
1120:
1118:
1117:
1112:
1110:
1109:
1093:
1091:
1090:
1085:
1065:
1063:
1062:
1057:
1045:
1043:
1042:
1037:
1025:
1023:
1022:
1017:
999:
997:
996:
991:
979:
977:
976:
971:
936:
934:
933:
928:
916:
914:
913:
908:
872:
870:
869:
864:
847:
846:
822:
821:
809:
808:
789:
787:
786:
781:
769:
767:
766:
761:
749:
747:
746:
741:
730:
729:
705:
704:
692:
691:
673:. Equivalently,
672:
670:
669:
664:
648:
646:
645:
640:
595:
593:
592:
587:
569:
567:
566:
563:{\displaystyle }
561:
543:
541:
540:
535:
523:
521:
520:
515:
503:
501:
500:
495:
483:
481:
480:
475:
451:
449:
448:
443:
425:
423:
422:
417:
406:also belongs to
405:
403:
402:
397:
392:
391:
367:
366:
354:
353:
334:
332:
331:
326:
302:
300:
299:
294:
282:
280:
279:
274:
262:
260:
259:
254:
249:
248:
224:
223:
211:
210:
191:
189:
188:
183:
181:
180:
161:
159:
158:
153:
141:
139:
138:
133:
131:
130:
111:
109:
108:
103:
75:such that every
74:
72:
71:
66:
53:finite sequences
50:
48:
47:
42:
1966:
1965:
1961:
1960:
1959:
1957:
1956:
1955:
1931:
1930:
1911:156. Springer.
1896:
1876:
1830:
1829:
1815:
1814:
1802:
1798:
1784:
1783:
1772:
1766:
1762:
1748:
1747:
1727:
1724:
1723:
1698:
1695:
1694:
1668:
1665:
1664:
1641:
1637:
1619:
1615:
1610:
1607:
1606:
1586:
1582:
1570:
1566:
1564:
1561:
1560:
1531:
1527:
1515:
1511:
1499:
1495:
1486:
1482:
1473:
1469:
1460:
1456:
1451:
1448:
1447:
1431:
1428:
1427:
1411:
1408:
1407:
1387:
1383:
1369:
1366:
1365:
1349:
1346:
1345:
1323:
1320:
1319:
1297:
1294:
1293:
1271:
1268:
1267:
1251:
1248:
1247:
1231:
1228:
1227:
1211:
1208:
1207:
1185:
1182:
1181:
1180:is of the form
1164:
1160:
1158:
1155:
1154:
1138:
1135:
1134:
1105:
1101:
1099:
1096:
1095:
1079:
1076:
1075:
1072:
1051:
1048:
1047:
1031:
1028:
1027:
1026:if and only if
1005:
1002:
1001:
1000:are ordered by
985:
982:
981:
965:
962:
961:
954:minimal element
922:
919:
918:
902:
899:
898:
883:
836:
832:
817:
813:
804:
800:
795:
792:
791:
790:such that that
775:
772:
771:
755:
752:
751:
719:
715:
700:
696:
687:
683:
678:
675:
674:
658:
655:
654:
634:
631:
630:
627:
581:
578:
577:
570:and called the
549:
546:
545:
529:
526:
525:
509:
506:
505:
489:
486:
485:
469:
466:
465:
464:through a tree
458:
431:
428:
427:
411:
408:
407:
381:
377:
362:
358:
349:
345:
340:
337:
336:
308:
305:
304:
288:
285:
284:
268:
265:
264:
238:
234:
219:
215:
206:
202:
197:
194:
193:
192:, such that if
173:
169:
167:
164:
163:
147:
144:
143:
123:
119:
117:
114:
113:
97:
94:
93:
90:
85:
60:
57:
56:
55:of elements of
36:
33:
32:
21:
12:
11:
5:
1964:
1954:
1953:
1948:
1943:
1929:
1928:
1895:
1892:
1891:
1890:
1875:
1872:
1871:
1870:
1858:
1855:
1852:
1849:
1846:
1843:
1837:
1834:
1828:
1822:
1819:
1813:
1810:
1805:
1801:
1797:
1791:
1788:
1782:
1779:
1775:
1769:
1765:
1761:
1755:
1752:
1746:
1743:
1740:
1737:
1734:
1731:
1708:
1705:
1702:
1678:
1675:
1672:
1652:
1647:
1644:
1640:
1636:
1633:
1630:
1625:
1622:
1618:
1614:
1592:
1589:
1585:
1581:
1576:
1573:
1569:
1548:
1543:
1540:
1537:
1534:
1530:
1526:
1521:
1518:
1514:
1510:
1507:
1502:
1498:
1494:
1489:
1485:
1481:
1476:
1472:
1468:
1463:
1459:
1455:
1435:
1415:
1393:
1390:
1386:
1382:
1379:
1376:
1373:
1353:
1333:
1330:
1327:
1301:
1292:of every tree
1281:
1278:
1275:
1255:
1235:
1226:. Namely, let
1215:
1195:
1192:
1189:
1167:
1163:
1142:
1131:discrete space
1108:
1104:
1083:
1071:
1068:
1055:
1035:
1015:
1012:
1009:
989:
969:
926:
906:
895:directed graph
882:
879:
862:
859:
856:
853:
850:
845:
842:
839:
835:
831:
828:
825:
820:
816:
812:
807:
803:
799:
779:
759:
739:
736:
733:
728:
725:
722:
718:
714:
711:
708:
703:
699:
695:
690:
686:
682:
662:
638:
626:
625:Terminal nodes
623:
617:, a tree on a
585:
559:
556:
553:
533:
513:
493:
473:
457:
454:
441:
438:
435:
415:
395:
390:
387:
384:
380:
376:
373:
370:
365:
361:
357:
352:
348:
344:
324:
321:
318:
315:
312:
292:
272:
252:
247:
244:
241:
237:
233:
230:
227:
222:
218:
214:
209:
205:
201:
179:
176:
172:
151:
129:
126:
122:
101:
89:
86:
84:
81:
64:
40:
9:
6:
4:
3:
2:
1963:
1952:
1949:
1947:
1944:
1942:
1939:
1938:
1936:
1925:
1924:3-540-94374-9
1921:
1918:
1917:0-387-94374-9
1914:
1910:
1906:
1902:
1898:
1897:
1889:
1885:
1881:
1878:
1877:
1850:
1844:
1832:
1826:
1817:
1803:
1799:
1795:
1786:
1767:
1763:
1759:
1750:
1741:
1735:
1729:
1722:
1721:
1720:
1703:
1692:
1673:
1645:
1642:
1638:
1631:
1623:
1620:
1616:
1590:
1587:
1583:
1579:
1574:
1571:
1567:
1541:
1538:
1535:
1532:
1528:
1524:
1519:
1516:
1512:
1508:
1505:
1500:
1496:
1492:
1487:
1483:
1479:
1474:
1470:
1466:
1461:
1457:
1433:
1413:
1391:
1388:
1380:
1377:
1374:
1351:
1331:
1328:
1325:
1318:
1313:
1299:
1276:
1253:
1233:
1213:
1190:
1165:
1161:
1140:
1132:
1128:
1124:
1106:
1102:
1081:
1067:
1053:
1033:
1013:
1010:
1007:
987:
967:
959:
955:
951:
947:
943:
938:
924:
904:
896:
892:
888:
878:
876:
860:
857:
851:
848:
843:
840:
837:
833:
829:
826:
823:
818:
814:
810:
805:
801:
777:
757:
737:
734:
726:
723:
720:
716:
712:
709:
706:
701:
697:
693:
688:
684:
660:
652:
651:terminal node
636:
622:
620:
616:
615:Kőnig's lemma
612:
611:
606:
605:
604:
597:
583:
575:
574:
554:
531:
511:
491:
471:
463:
453:
439:
436:
433:
413:
388:
385:
382:
378:
374:
371:
368:
363:
359:
355:
350:
346:
322:
319:
316:
313:
310:
290:
270:
245:
242:
239:
235:
231:
228:
225:
220:
216:
212:
207:
203:
177:
174:
170:
149:
127:
124:
120:
99:
80:
78:
62:
54:
38:
30:
26:
19:
1904:
1690:
1314:
1126:
1094:(denoted as
1073:
958:well-ordered
942:order theory
939:
887:graph theory
884:
874:
650:
649:is called a
628:
609:
608:
601:
600:
598:
576:of the tree
572:
571:
461:
459:
91:
28:
22:
1951:Determinacy
1125:, treating
891:rooted tree
603:wellfounded
544:is denoted
112:is denoted
83:Definitions
1935:Categories
1894:References
1884:set theory
1880:Laver tree
1691:projection
619:finite set
610:illfounded
1845:∈
1842:⟩
1836:→
1821:→
1812:⟨
1804:ω
1796:∈
1790:→
1781:∃
1768:ω
1760:∈
1754:→
1646:ω
1632:×
1624:ω
1591:ω
1580:×
1575:ω
1547:⟩
1506:…
1454:⟨
1392:ω
1378:×
1329:×
1166:ω
1107:ω
952:with one
858:∈
855:⟩
841:−
827:…
798:⟨
735:∈
732:⟩
724:−
710:…
681:⟨
394:⟩
386:−
372:…
343:⟨
314:≤
303:, and if
251:⟩
243:−
229:…
200:⟨
178:ω
128:ω
31:on a set
1903:(1995).
1874:See also
1070:Topology
1888:forcing
1446:(e.g.,
1922:
1915:
875:pruned
613:. By
462:branch
77:prefix
1663:with
1129:as a
948:is a
893:is a
88:Trees
1920:ISBN
1913:ISBN
1643:<
1621:<
1588:<
1572:<
1389:<
1011:<
980:and
889:, a
573:body
320:<
175:<
125:<
29:tree
27:, a
1693:of
1153:of
940:In
885:In
770:of
283:in
162:of
23:In
1937::
1907:.
1719:,
877:.
596:.
460:A
1926:.
1869:.
1857:}
1854:]
1851:T
1848:[
1833:y
1827:,
1818:x
1809:)
1800:Y
1787:y
1778:(
1774:|
1764:X
1751:x
1745:{
1742:=
1739:]
1736:T
1733:[
1730:p
1707:]
1704:T
1701:[
1677:]
1674:T
1671:[
1651:]
1639:Y
1635:[
1629:]
1617:X
1613:[
1584:Y
1568:X
1542:1
1539:+
1536:m
1533:2
1529:y
1525:,
1520:m
1517:2
1513:x
1509:,
1501:3
1497:y
1493:,
1488:2
1484:x
1480:,
1475:1
1471:y
1467:,
1462:0
1458:x
1434:Y
1414:X
1385:)
1381:Y
1375:X
1372:(
1352:T
1332:Y
1326:X
1300:T
1280:]
1277:T
1274:[
1254:C
1234:T
1214:T
1194:]
1191:T
1188:[
1162:X
1141:C
1127:X
1103:X
1082:X
1054:U
1034:T
1014:U
1008:T
988:U
968:T
925:T
905:T
861:T
852:x
849:,
844:1
838:n
834:x
830:,
824:,
819:1
815:x
811:,
806:0
802:x
778:X
758:x
738:T
727:1
721:n
717:x
713:,
707:,
702:1
698:x
694:,
689:0
685:x
661:T
637:T
584:T
558:]
555:T
552:[
532:T
512:T
492:X
472:T
440:0
437:=
434:m
414:T
389:1
383:m
379:x
375:,
369:,
364:1
360:x
356:,
351:0
347:x
323:n
317:m
311:0
291:T
271:n
246:1
240:n
236:x
232:,
226:,
221:1
217:x
213:,
208:0
204:x
171:X
150:T
121:X
100:X
63:X
39:X
20:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.