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:

Example 1:AdditionSuppose 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?