There are two principal ideas to recognize in this example. The first is that binomial trees can be calibrated to the real world by using the Cox-Ross-Rubinstein approach. The second relates to the valuation of an American option - at each node prior to maturity a check needs to be made for early exercise.

Use the Cox-Ross-Rubinstein approach to value the American put option using a six-step tree.
Calculate t, u, d, p, and 1 - p for the six-step tree.
Number of time steps					6
Length of each time step (in years)				t
Up multiplier				u
Down multiplier				d
Risk-neutral probability of up movement				p
Risk-neutral probability of down movement				1 - p
1) Build the stock price tree
2) Calculate the payoff of the option at each terminal node
3) Use backwards induction to find the discounted expected payoffs at intermediate nodes, checking for early exercise at each node