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A356917 Irregular triangle read by rows where row n lists the Colijn-Plazzotta subtree numbers, in ascending order, of each vertex of the rooted binary tree with their tree number n.

%I A356917 #19 Dec 19 2024 11:46:19
%S A356917 1,1,1,2,1,1,1,2,3,1,1,1,1,2,2,4,1,1,1,1,2,3,5,1,1,1,1,1,2,2,3,6,1,1,
%T A356917 1,1,1,1,2,2,3,3,7,1,1,1,1,1,2,2,4,8,1,1,1,1,1,1,2,2,2,4,9,1,1,1,1,1,
%U A356917 1,1,2,2,2,3,4,10,1,1,1,1,1,1,1,1,2,2,2,2,4,4,11
%N A356917 Irregular triangle read by rows where row n lists the Colijn-Plazzotta subtree numbers, in ascending order, of each vertex of the rooted binary tree with their tree number n.
%C A356917 Colijn and Plazzotta enumerate rooted binary trees (every vertex 0 or 2 children) by n=1 as a singleton and thereafter tree n is a root with child subtrees x = A002024(n-1) and y = A002260(n-1).
%C A356917 Each row starts with 1's for the childless vertices (A064064(n) of them).
%C A356917 Each row ends with n itself (the tree root).
%C A356917 The second last term in each row is the numerically largest subtree of the root, which is x.
%C A356917 Row lengths are A064002(n), the number of vertices.
%H A356917 Kevin Ryde, <a href="/A356917/b356917.txt">Table of n, a(n) for rows 1..500, flattened</a>
%H A356917 Caroline Colijn and Giacomo Plazzotta, <a href="https://doi.org/10.1093/sysbio/syx046">A Metric on Phylogenetic Tree Shapes</a>, Systematic Biology, volume 67, number 1, January 2018, pages 113-126, see section 2.3 where their L_n = row n here.
%H A356917 Kevin Ryde, <a href="/A356917/a356917.gp.txt">PARI/GP Code</a>
%F A356917 row(n) = sort {row(x), row(y), n} for n>=2, where x = A002024(n-1) and y = A002260(n-1).
%e A356917 Triangle begins:
%e A356917       k=1  2  3  4  5  6  7  8  9 10 11
%e A356917   n=1:  1,
%e A356917   n=2:  1, 1, 2,
%e A356917   n=3:  1, 1, 1, 2, 3,
%e A356917   n=4:  1, 1, 1, 1, 2, 2, 4,
%e A356917   n=5:  1, 1, 1, 1, 2, 3, 5,
%e A356917   n=6:  1, 1, 1, 1, 1, 2, 2, 3, 6,
%e A356917   n=7:  1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 7,
%e A356917   n=8:  1, 1, 1, 1, 1, 2, 2, 4, 8,
%e A356917 Tree n=6 and its subtree numbers are as follows and row 6 is these subtree numbers in ascending order.
%e A356917           6  root
%e A356917         /   \
%e A356917       3       2
%e A356917      / \     / \
%e A356917     2   1   1   1
%e A356917    / \
%e A356917   1   1
%o A356917 (PARI) \\ See links.
%Y A356917 Cf. A002024, A002260 (root subtrees).
%Y A356917 Cf. A064002 (number of vertices), A064064 (number of childless).
%Y A356917 Cf. A356918 (d1 metric).
%K A356917 nonn,easy,tabf
%O A356917 1,4
%A A356917 _Kevin Ryde_, Sep 19 2022