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A228155 Maximal external path length of AVL trees with n (leaf-) nodes. 3
0, 2, 5, 8, 12, 16, 20, 25, 30, 35, 40, 44, 50, 56, 62, 68, 73, 79, 85, 91, 97, 103, 110, 117, 123, 130, 137, 144, 151, 157, 163, 170, 177, 184, 191, 197, 204, 211, 219, 227, 235, 243, 250, 257, 265, 273, 281, 289, 296, 304, 312, 320, 328, 335, 342, 349, 356 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The external path length of a tree is the sum of the levels of its external nodes (i.e. leaves).
REFERENCES
D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 6.2.3 (7) and (8).
LINKS
R. C. Richards, Shape distribution of height-balanced trees, Info. Proc. Lett., 17 (1983), 17-20.
Wikipedia, AVL tree
EXAMPLE
The (two) AVL trees with 3 (leaf-) nodes have one with depth 1 and two with depth 2:
o o
/ \ / \
o 1 1 o
/ \ / \
2 2 2 2
so a(3) = 5.
MATHEMATICA
maxNods = 100; Clear[T, A029837, A072649, A036289, A228155]; T[0, 1] = 0; A029837[1] = 0; A072649[1] = 1; A228155[1] = 0; For[k = 2, k <= maxNods, k++, A029837[k] = maxNods; A072649[k] = 0; A228155u = 0; For[kL = 1, kL <= Floor[k/2], kL++, For[hL = A029837[kL], hL <= A072649[kL] - 1, hL++, For[hR = Max[hL - 1, A029837[k - kL]], hR <= Min[hL + 1, A072649[k - kL] - 1], hR++, maxDepthSum = k + T[hL, kL] + T[hR, k - kL]; A228155u = Max[maxDepthSum, A228155u]; h = Max[hL, hR] + 1; If[ !IntegerQ[T[h, k]], T[h, k] = maxDepthSum, T[h, k] = Max[maxDepthSum, T[h, k]]]; A029837[k] = Min[h, A029837[k]]; If[ !IntegerQ[A036289[h]], A036289[h] = maxDepthSum, A036289[h] = Max[maxDepthSum, A036289[h]]]; A072649[k] = Max[h + 1, A072649[k]]; ]]]; A228155[k] = A228155u]; k =.; Table[A228155[k], {k, 1, maxNods}] (* Jean-François Alcover, Aug 19 2013, translated and adapted from Herbert Eberle's MuPAD program *)
PROG
(MuPAD)
maxNods:=100: // max number of leaves (= external nodes)
// Triangle T for all AVL trees with <= maxNods leaves:
delete T:
// table T indexed [h, k] (h=height, k=number of leaves):
T[0, 1]:=0:
// A029837 indexed [k], min height of tree with k leaves:
A029837:=array(1..maxNods): A029837[1]:=0:
// A072649 indexed [k], 1+max height of AVL tree with k leaves:
A072649:=array(1..maxNods): A072649[1]:=1:
// A036289 indexed [h], max depthsum of all height h AVL trees:
A036289:=array(1..maxNods):
// A228155 indexed [k], max depthsum of all AVL trees with k leaves:
A228155:=array(1..maxNods): A228155[1]:=0:
for k from 2 to maxNods do:
A029837[k]:=maxNods: // try infinity for the min height
A072649[k]:=0:
A228155u:=0:
// Put together 2 AVL trees:
for kL from 1 to floor(k/2) do:
// kL leaves in the left tree
for hL from A029837[kL] to A072649[kL]-1 do:
for hR from max(hL-1, A029837[k-kL])
to min(hL+1, A072649[k-kL]-1) do:
// k-kL leaves in the right subtree
maxDepthSum:=T[hL, kL]+T[hR, k-kL]+k:
A228155u:=max(maxDepthSum, A228155u):
h:=max(hL, hR)+1:
if type(T[h, k]) <> DOM_INT then // T[h, k] uninit
T[h, k]:=maxDepthSum:
else
T[h, k]:=max(maxDepthSum, T[h, k]):
end_if:
A029837[k]:=min(h, A029837[k]):
if type(A036289[h]) <> DOM_INT then
A036289[h]:=maxDepthSum:
else
A036289[h]:=max(maxDepthSum, A036289[h]):
end_if:
A072649[k]:=max(h+1, A072649[k]):
end_for: // hR
end_for: // hL
end_for: // kL
A228155[k]:=A228155u:
end_for: // k
CROSSREFS
Column maxima of triangles A228152, A228153.
Row maxima give: n*2^n = A036289(n).
Row minima give: A067331(n-1) for n>0 or A166106(n+2).
Row lengths give: 1+A008466(n).
Column heights give: A217710(k).
Number of AVL trees read by rows gives: A143897.
The infimum of all external path lengths of all binary trees with k (leaf-) nodes is: A003314(k) for k>0.
Number of AVL trees read by columns gives: A217298.
Sequence in context: A228152 A061717 A003314 * A070977 A134925 A184430
KEYWORD
nonn
AUTHOR
Herbert Eberle, Aug 14 2013
STATUS
approved

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Last modified March 28 11:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)