# Can’t understand this example of convergecast addition from a book

In the book Distributed Computing: A Locality-Sensitive Approach by David Peleg, in chapter 3, Broadcast and convergecast, in section 3.4.2 Global function computation, examples are given for global computation of functions. I can’t understand the first example, of addition:

Suppose that the function $$f$$ represents addition. Note that if the inputs are $$m$$-bit integers, then $$f\left(\mathcal{Y}\right)$$ can be represented in $$O\left(m\log{n}\right)$$ bits for any $$\mathcal{Y}\subseteq\mathcal{X}$$. Therfore, the message and time cmplexities of the convergecast process performed by Procedure CONVERGE($$+$$) on the tree $$T$$ are $$O\left(nm\right)$$ and $$O\left(Depth\left(T\right){\cdot}m\right)$$, respectivly.
What I don’t understand is why the complexities are $$O\left(nm\right)$$ and $$O\left(Depth\left(T\right){\cdot}m\right)$$ and not just the “normal” $$O\left(n\right)$$ and $$O\left(Depth\left(T\right)\right)$$. I realize that this is comehow related the $$m$$-bit representation of the integers iputs. Why is there a factor $$m$$ in the complexities?