Answer
See the explanation
Work Step by Step
a) In a binary tree, each node can produce 2 children. If the goal is reached after 8 productions, that means the goal is at depth 8 (starting from depth 0).
Breadth-First Search (BFS)
BFS explores all nodes at depth 0, then depth 1, then depth 2, and so on—layer by layer.
Total Nodes Up to Depth 8
The number of nodes in a full binary tree up to depth
𝑑 is:
Total nodes $= \sum_{i=0}^d 2^i=2^{d+1}-1$
For $d=8$
Total nodes $= 2^9-1=512-1=511$
The maximum number of nodes in a binary tree with depth 8, constructed using a breadth-first manner is 511.
b) Instead of searching from the start all the way to the goal, bidirectional search starts:
One search from the initial state
One search from the goal state
Both searches proceed simultaneously until they meet in the middle
Why It’s Efficient:
Each search only needs to explore half the depth.
If the goal is at depth
𝑑, each search goes to depth
𝑑/2.
Node Reduction in Binary Tree
Using the same formula:
Depth $𝑑=8$
Each search explores up to depth 4
Nodes per search $=2^5-1=31$
Total nodes $= 31+31=62$
Compare that to 511 nodes in full BFS—this is a dramatic reduction.
So bidirectional search reduces the number of nodes considered from 511 to just 62 in this case—by cutting the search depth in half and leveraging symmetry.
