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Bhathiya edited section_Challenegs_in_Bushy_Trees__.tex
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\section{Challenegs in Bushy Tree Implementation}
Moving from left-deep tree to a Bushy tree is a challenge as the number of possible structures in bushy trees are much larger. Left-deep trees have only one structure regardless of the number of attributes involved. Therefor the number of possible permutations are N! . But for bushy trees possible number of structures are given by
$$f(x) $$S(N) = \begin{cases}
0 & 1 \text{if
$0 < x \le 0.05$}; \\
0.1 & \text{if $0.05 < x \le $ N = 1$}; \\
0.2 & $\displaystyle \sum_{i=1}^{N} \frac{1}{n}$ \text{if
$1 < x \le 5$};\\
2^{\frac{x}{20}} & $N \ne 0$}
2 \text{if
$5 < x \le 100$};.\end{cases} $$ $ N = 1$}.\end{cases}$$
Therefore the number of possible permutations are $S(N)*N!$. Unlike left-deep tree case, estimating the cost for all the possible bushy trees is computationally infeasible for moderately large N. Therefore it is required to come with a heuristic to select set of permutations for cost computations.Some heuristics considered are