0,1

This sequence is based on observation that the terms of A014486 (2n-digit balanced binary sequences) encode rooted plane trees with n+1 vertices (n edges), but also rooted binary plane trees with n+1 leaves, i.e. 2n edges, 2n+1 vertices.

Index entries for sequences related to rooted trees

We start from the simplest such binary tree: 0.0 (binary depth-first encoding = 2, from left-to-right, 1 with the zero of the last leaf ignored); then encode it as an ordinary rooted plane tree (depth-first wise) to get the code 1010 = decimal 10, which in turn, when interpreted as an encoding of binary tree is:

..0.0

.0.1. (whose rooted plane tree coding is 10110100 = 180 in decimal)

..1.. etc.

a(n) = bt_df2tree_apply_k_times(2, n)

bt_df2tree_apply_k_times := proc(n, k) option remember; if(0 = k) then (n) else bt_df2tree_apply_k_times(bintree_depth_first2tree(n), k-1); fi; end;

bintree_depth_first2tree := n -> ((btdf2t(n*2, floor_log_2(n)+1)/2) - 2^(2*(floor_log_2(n)+1)));

btdf2t := proc(n, ii) local i, e, x, y; i := ii; if(n >= (2^i)) then x := btdf2t(n - (2^i), i-1); i := i - ((floor_log_2(x)+1)/2); y := btdf2t((n mod (2^i)), i-1); RETURN((2^(floor_log_2(y)+2))*((2^(floor_log_2(x)+1)) + x) + 2*y); else RETURN(2); fi; end;

Cf. A057120, A057121, A057122.

Sequence in context: A069994 A063573 A086675 this_sequence A037267 A155200 A156510

Adjacent sequences: A057116 A057117 A057118 this_sequence A057120 A057121 A057122

nonn

Antti Karttunen Aug 11 2000