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#Htere is no tree it is said how to#

Though the fig tree does not blossom And there is no fruit on the vines, Though the yield of the olive fails And the fields produce no food, Though the flock is cut off from the fold And there are no cattle in the stalls, Though the fig tree should not blossom, And there be no fruit on the vines, Though the yield of the olive should fail, And the fields produce no food, Though the flock should be cut off from the fold, And there be no cattle in the stalls, Though the fig tree should not blossom And there be no fruit on the vines, Though the yield of the olive should fail And the fields produce no food, Though the flock should be cut off from the fold And there be no cattle in the stalls, Though the fig tree may not blossom, Nor fruit be on the vines Though the labor of the olive may fail, And the fields yield no food Though the flock may be cut off from the fold, And there be no herd in the stalls-Įven if the fig tree does not blossom, And there is no fruit on the vines, If the yield of the olive fails, And the fields produce no food, Even if the flock disappears from the fold, And there are no cattle in the stalls, Though the fig tree does not bud and no fruit is on the vines, though the olive crop fails and the fields produce no food, though the sheep are cut off from the fold and no cattle are in the stalls,Īlthough the fig tree shall not blossom, neither shall fruit be in the vines the labour of the olive shall fail, and the fields shall yield no meat the flock shall be cut off from the fold, and there shall be no herd in the stalls: Though the fig tree should not blossom, nor fruit be on the vines, the produce of the olive fail and the fields yield no food, the flock be cut off from the fold and there be no herd in the stalls, Though the fig tree does not bud and there are no grapes on the vines, though the olive crop fails and the fields produce no food, though there are no sheep in the pen and no cattle in the stalls,Įven though the fig trees have no blossoms, and there are no grapes on the vines even though the olive crop fails, and the fields lie empty and barren even though the flocks die in the fields, and the cattle barns are empty, Hence at level L, the number of vertices is:- * =.

Since we know that each vertex in a binary tree has the maximum of 2 vertices in next level, therefore the number of vertices on the level L is twice that of the level L-1. Therefore, maximum number of vertices on the level (L-1) is *** QuickLaTeX cannot compile formula: Now we assume that statement is true for the level (L-1). At level 0 (L=0), there is only one vertex at level (L=1), there is only *** QuickLaTeX cannot compile formula: Proof: The given theorem is proved with the help of mathematical induction.

Theorem 9: Prove the maximum number of vertices at level ‘L’ in a binary tree is, where L>=0.

If one edge is there then tree T has two centers. So in the end, if one vertex is there this implies tree T has one center. Note that all vertices that T had as centers will still remain centers in T’–>T”–>T”’–>….Ĭontinue this process until the remaining tree has either one vertex or one edge. Again delete pendant vertices from T’ so that resulting T” is still a tree with same centers. Delete all pendant vertices from T, then resulting graph T’ is still a tree. T must have at least two pendant vertices.

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#Htere is no tree it is said how to#

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