power instead of computing as we can reduce the number of multiplications if we computed ( ) instead better yetwhat if we just computed this is the idea behind recursive definition for raising value to an integer power (the expression / is integer division in which the real result is truncated if / ( xif is even / ( xif is odd listing provides recursive function for raising to the integer value of since two of the expressions compute ( ) / we go ahead and compute this value as the result on line and then determine if is even or odd if is eventhen the result is returnedotherwisewe have to first multiply the result by to include the odd factor the run time analysis of exp(is left as an exercise listing the recursive implementation of exp( recursive implementation for computing * where is an integer def expxn )if = return result expx xn / if = return result else return result playing tic-tac-toe in this technological ageit' very likely you have played computer game in which you are competing against the computer for exampleyou may have played the game of checkerschessor something as simple as tic-tac-toe in any such gamewhen it' the computer' turn to playthe computer must make decision as to what play to make depending on the game and your level of expertiseyou may sometimes think the computer is genius or that there is some kind of magic going on behind the scenes sohow does the computer make its decisionone simple technique the game programmer can apply is the use of game tree game tree provides the sequence of all possible moves that can be made in the game for both the computer and the human opponent when the computer has to make moveit can evaluate the game tree and determine its best move the best move in this case is one that allows the computer to win before its human opponent in the fewest possible moves thuswhen playing against computerit' not that the computer is highly intelligentbut that the computer can evaluate every possible move from the current point to the end of the game and choose the best move humans simply