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A356918 Triangle read by rows where T(n,k) is Colijn and Plazzotta's distance metric d_1(n,k) between rooted binary tree numbers n and k, for 1 <= k <= n.

%I A356918 #26 Dec 20 2024 10:45:21
%S A356918 0,2,0,4,2,0,6,4,4,0,6,4,2,4,0,8,6,4,4,4,0,10,8,6,6,6,4,0,8,6,6,2,6,4,
%T A356918 6,0,10,8,8,4,8,6,6,4,0,12,10,8,6,8,6,6,6,4,0,14,12,12,8,12,10,10,8,6,
%U A356918 6,0,8,6,4,6,2,4,6,6,8,8,12,0
%N A356918 Triangle read by rows where T(n,k) is Colijn and Plazzotta's distance metric d_1(n,k) between rooted binary tree numbers n and k, for 1 <= k <= n.
%C A356918 T(n,k) is the cardinality of the multiset symmetric difference ("XOR") between the subtree numbers in tree n, and in k, those being rows n and k of A356917.
%C A356918 A multiset symmetric difference discards copies of elements common to both sets, and keeps the excess copies which one of the multisets has over the other.
%C A356918 Equivalently, T(n,k) is the multi-dimensional Manhattan distance between vectors v_n and v_k where vector element v_t(s) is the number of occurrences of subtree number s in tree t.
%C A356918 Column k=1 it the distance to the singleton, which is a single subtree 1, so that T(n,1) = A064002(n) - 1 is the number of vertices of n except one 1.
%C A356918 The main diagonal is T(n,n) = 0 which is distance 0 between n and itself.
%C A356918 As a flat sequence, a(m) is distance d_1 between the two child subtrees of the root in tree number m+1.
%H A356918 Kevin Ryde, <a href="/A356918/b356918.txt">Table of n, a(n) for rows 1..150, flattened</a>
%H A356918 Caroline Colijn, <a href="https://github.com/carolinecolijn/treetop/">Treetop</a>,  R Code, see labeldistance() and distunlab().
%H A356918 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 d_1.
%H A356918 Kevin Ryde, <a href="/A356918/a356918.gp.txt">PARI/GP Code</a>
%F A356918 T(n,k) = Sum_{s = subtree numbers in n or k} abs(v_n(s) - v_k(s)) where v_t(s) is the number of times s occurs in row t of A356917.
%e A356918 Triangle begins:
%e A356918        k=1  2  3  4  5  6  7  8
%e A356918   n=1:   0,
%e A356918   n=2:   2, 0,
%e A356918   n=3:   4, 2, 0,
%e A356918   n=4:   6, 4, 4, 0,
%e A356918   n=5:   6, 4, 2, 4, 0,
%e A356918   n=6:   8, 6, 4, 4, 4, 0,
%e A356918   n=7:  10, 8, 6, 6, 6, 4, 0,
%e A356918   n=8:   8, 6, 6, 2, 6, 4, 6, 0,
%e A356918   ...
%e A356918 For n=68,k=4, rows 68 and 4 from A356917 are as follows and their multiset symmetric difference has T(68,4) = 8 terms.
%e A356918   n=68:  1,1,1,1,1,1, 2,   3,    5,12,68
%e A356918   k= 4:  1,1,1,1,     2,2,    4
%e A356918   diff:          1,1,   2, 3, 4, 5,12,68
%o A356918 (PARI) \\ See links.
%o A356918 (R) # See links.
%Y A356918 Cf. A356917 (subtree numbers).
%Y A356918 Cf. A002024, A002260 (root subtrees).
%Y A356918 Cf. A064002 (number of vertices).
%K A356918 nonn,tabl
%O A356918 1,2
%A A356918 _Kevin Ryde_, Sep 19 2022