Consider a process $H$ where, starting from the value $h=0$, an amount of $x^{-n}$ is added to $h$ with probability $P$, an amount of $x^{-n}$ is subtracted from $h$ with probability $Q$, and nothing happens with probability $1-P-Q$, for every generation $n$, where $x,n\in\mathbb{N}$ and $x>2$, $n\geq1$.
One can write this process as a ternary tree where the leaves for a generation $n$ are the discrete values of $h\in[-\frac{1-x^{-n}}{x-1},\frac{1-x^{-n}}{x-1}]$.
I am trying to find the sum $$ \sum\limits_{h'\geq h} P^{(n)}(H = h')\,, $$ where $P^{(n)}(H = h')$ is the probability that after $n$ generations, the height has value $h'$. The sum of course stops at $h' = \frac{1-x^{-n}}{x-1}$.
The problem is that the $h'$ are not equally spaced and should be summed according to their increasing value. Is there a simple expression for this sum?