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A search algorithm is complete for a specified class of problems if it finds a solution whenever a solution exists in that class. This definition need not require termination on unsolvable instances; an additional termination guarantee should be stated separately. A finite branching factor makes breadth-first tree search complete for finite-depth solutions, while depth-first tree search additionally needs a reason it cannot be trapped on an infinite branch.

A tree in which every vertex has finitely many incident edges; for a rooted tree this is equivalent to finitely many children per node. Induction on depth then shows every level from a single root is finite, even if the entire tree is infinite. Consequently ordinary breadth-first tree search can eventually reach each node, provided the elementary traversal operations terminate.

A search algorithm is optimal for a specified objective if its returned solution has the best objective value among all solutions. The objective is part of the claim: minimizing the number of actions does not generally maximize a reward assigned to terminal states. Breadth-first tree search optimizes total cost when all actions have the same positive cost.

Breadth-first tree search is complete and optimal for minimizing rounds. If a successful escape has finite length $d$, there are only finitely many nodes at depths below $d$, since the branching factor is at most five. Breadth-first tree search therefore reaches a shallowest successful node after finite work, even if other branches continue forever.

It is an optimal search algorithm because every search transition model step costs one round, so minimum depth is exactly the desired minimum number of rounds. The goal test must be applied to the surviving complete-round successor. Here a complete search algorithm guarantees finding an escape when one exists; it need not terminate on every unsolvable instance.

Breadth-first tree search remains complete but is not optimal for maximizing the bonus. Completeness depends on the finite branching factor and a finite successful depth, not on the objective used to compare escapes.

Use the one-row, seven-square example from part (e). The one-round escape at column $3$ leaves the fox at column $2$, so its graph distance to the exit, and hence the bonus, is $1$. The two-round escape at column $6$ leaves the fox at column $4$, giving bonus $2$. The earliest successful node therefore has bonus $1$ when bonus $2$ is achievable. Breadth-first tree search orders by rounds rather than terminal graph distance.

Breadth-first tree search is complete and is still not bonus-optimal. Fix any deterministic rule for choosing among equally short fox routes; the completeness argument again uses only the finite branching factor. The unobstructed corridor of part (g) has unique shortest paths, so its fox moves are exactly the same under the new strategy. The one-round bonus is $1$ and the two-round bonus is $2$, proving nonoptimality without any tie-breaking ambiguity.

If there is no route between fox and rabbit, the printed shortest-path rule needs a convention; letting the fox stay supplies a natural extension. The completeness argument is unchanged, and the counterexample above uses only connected squares. When the fox and rabbit initially belong to the same connected component of a graph, the shortest-path pursuit distance bound also makes every surviving branch finite. That additional property does not make the first escape maximize its bonus.