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In game theory, terminal vertices are the end points of a game tree, representing specific outcomes or final states in a game. They are where the game concludes, and at these points, the results are concrete with no further moves possible. This is crucial in evaluating a game tree because the values assigned to terminal vertices are the base for determining game strategies and predicting outcomes.

For instance, consider a simple game where two players pick numbers in turn from a finite set with the goal of maximizing their score. The terminal vertices occur when all numbers are picked. If these endpoints have values assigned based on the final score of each player, such as +1 for a win, 0 for a draw, and -1 for a loss, it gives us the definitive outcomes to work backward from to understand the game's dynamics.

Backward induction is a method used in game theory to solve a game from the end to the beginning. By beginning at the terminal vertices, one can reason backwards to infer the optimal moves for each player at every step of the game. This backward analysis provides insight on the choices that both players should make to optimize their outcomes at each stage.

During evaluation, you'd look at the known values of terminal vertices and determine what the previous moves must have been for rational, winning players. At each non-terminal vertex one level above, you assign the value based on the optimal choice available. For example, if a non-terminal vertex connects to terminal vertices with values 3 and 5, and it's the turn of the player who benefits from higher scores, you'd assign it a value of 5. Continue this process all the way up to the root of the tree to deduce the course of optimal play from start to finish.

Optimal play in the context of game trees refers to making the best possible moves at every point in the game. It is a strategy that, when followed by rational and perfectly informed players, leads to outcomes that maximize their gains or minimize their losses. In the steps of game tree evaluation, once we reach the root after applying backward induction, we have a value that represents the game's outcome assuming optimal play by all players.

In a turn-based game, this means that players are expected to choose the move leading to the highest value at each decision point. Understanding optimal play is critical not just for predicting game outcomes, but also for game design, creating challenges that are balanced and strategically complex.