deletions | additions       diff --git a/section_Challenegs_in_Bushy_Trees__.tex b/section_Challenegs_in_Bushy_Trees__.tex  index 98292b5..1b4306b 100644  --- a/section_Challenegs_in_Bushy_Trees__.tex  +++ b/section_Challenegs_in_Bushy_Trees__.tex 
...
\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 & \text{if $0 < x \le 0.05$}; \\  0.1 & \text{if $0.05 < x \le 1$}; 1 if $N = 1 $  \\ 0.2 & \text{if $1 < x \le 5$};\\  2^{\frac{x}{20}} & \text{if $5 < x \le 100$};.\end{cases} $$ S(N) = $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n}$ if N > 1 $$\\  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