Identifier
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=>
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[[[[[[],[]],[]],[[],[]]],[]],[]]=>[[[[[.,.],[.,.]],[.,.]],[.,[[.,.],[.,.]]]],[.,.]]
[[[[[],[[],[[],[]]]],[]],[]],[]]=>[[[[.,[[.,[[.,[.,.]],[.,.]]],.]],.],[.,.]],[.,.]]
[[[[[],[[[],[]],[]]],[]],[]],[]]=>[[[[.,[[[.,[.,.]],[.,.]],[.,.]]],.],[.,.]],[.,.]]
[[[[[[],[]],[[],[]]],[]],[]],[]]=>[[[[[.,.],[.,.]],[.,[[.,.],[.,.]]]],[.,.]],[.,.]]
[[[[[[],[[],[]]],[]],[]],[]],[]]=>[[[[[.,[[.,[.,.]],[.,.]]],.],[.,.]],[.,.]],[.,.]]
[[[[[[[],[]],[]],[]],[]],[]],[]]=>[[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]
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(only identities of compositions of up to three maps are shown)
Description
Zeilberger's Strahler bijection between ordered and binary trees.
This is a bijection sending the pruning number of the ordered tree to the Strahler number of the binary tree.
This is a bijection sending the pruning number of the ordered tree to the Strahler number of the binary tree.
References
[1] Zeilberger, D. A bijection from ordered trees to binary trees that sends the pruning order to the Strahler number MathSciNet:1058712
Properties
bijective, graded
Sage code
def mapping(t): """ sage: mapping([]) . sage: mapping([[]]) [., .] sage: [mapping(t) for t in OrderedTrees(3)] [[., [., .]], [[., .], .]] """ def prune(t): def is_branch(b): n = len(b) return n == 0 or (n == 1 and is_branch(b[0])) if len(t) == 0: return [] else: return [prune(b) for b in t if not is_branch(b) ] def pruning_order(t): o = 0 while len(t) > 0: t = prune(t) o += 1 return o def below(t): t1 = t[0] t2 = t[1:] k = pruning_order(t) k1 = pruning_order(t1) k2 = pruning_order(t2) if (k1 <= k-1 and k2 == k) or (k1 == k-1 and k2 == k-1): return (t1, t2) elif k1 == k-1 and k2 <= k-2: i = 0 while pruning_order(t1[:i+1]) <= k-2: i += 1 f1 = t1[:i] s1 = t1[i] j = i+1 while pruning_order(t1[i+1:j+1]) <= k-2: j += 1 f2 = t1[i+1:j] return (f1+[s1]+f2, t2+t1[j:]) elif k1 == k: i = 0 while pruning_order(t1[:i+1]) <= k-1: i += 1 f1 = t1[:i] s1 = t1[i] f2 = t1[i+1:] return (f1+[s1]+t2, f2) if len(t) == 0: return BinaryTree() else: u1, u2 = below(t) return BinaryTree([mapping(u1), mapping(u2)])
Weight
26
Created
Jan 29, 2020 at 13:27 by FindStatCrew
Updated
Jan 29, 2020 at 13:27 by Martin Rubey
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