"A two person games begins with the random selection of an integer $x$ from the set {$1,2,3$}, each choice is equally likely. Then the two players, not knowing the value of $x$, simultaneously select integers from {$1,2,3$}. Each players objective is to choose an integer $\geq x$. If P1 is successful, he wins, if both are successful then the player who chose the smallest integer wins, in all other cases it is a draw. The winner gets one dollar from the loser."
I have drawn my tree for this, 
But I am really struggling to understand how I find my pure strategy solution for each player and then the pay off matrix?
You already did the hardest part, now write 3 payoff matrices. The first matrix $M1$ corresponds to the game if Nature chose 1: $$ \begin{array}{c|c|c|c|} P1\backslash P2\\\hline\\& 1 & 2 &3\\ \hline 1 & 0 & 1 & 1\\ \hline 2 &-1 &0 &1 \\ \hline 3 & -1 &-1 &0 \\ \hline\end{array} \tag{M1}$$ The second matrix $M2$ corresponds to the game if Nature chose 2, and the third $M3$ if Nature chose 3. Of course, none of these matrices makes sense since players do not observe Nature's choice. But now, computed the expected matrix ( $\frac{1}{3} M_1+\frac 13 M_2 + \frac 13 M_3$), using the expected matrix you can compute the Nash equilibrium (or equilibria) as usual.