deletions | additions       diff --git a/section_Query_Optimization_and_Execution__.tex b/section_Query_Optimization_and_Execution__.tex  index 3f1ac71..0548c44 100644  --- a/section_Query_Optimization_and_Execution__.tex  +++ b/section_Query_Optimization_and_Execution__.tex 
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Two possible execution plans are deep-left tree (shown in Figure-1 ) and Bushy Tree (shown in Figure-2). Other possible structures include right-deep trees zig-zag trees. Similar to deep-left trees right-deep and zig-zag are linear tree structure. But bushy trees are different and more complicated compared to others.   \subsection{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 tables involved. Therefor the number of possible permutations are N! . But for bushy trees possible number of structures are given by \\  $$ S(N) = 1 if N = 1\\   S(N) = $\displaystyle \sum_{N=1}^{i} (N)(N-i)$ if N > 1 $$\\