Error bound for minimum-degree spanning tree

I am reading The Design of Approximation Algorithms (Williamson and Schmoys; page 50 of this PDF) about solving the minimum-degree spanning tree problem of maximum degree 2 in polynomial time. The text states that we are going to find a tree $T$ with maximum degree at most $2\mathrm{OPT} +\lceil\log_{2} n\rceil$ in polynomial-time. $n$ is the number of the vertices in the graph. Could someone clarify why $\log_{2} n$ is placed there? The text gives no explanation what it is.

It is $\sf NP$-complete to decide if a given a graph has a minimum-degree spanning tree of maximum degree 2. Because the problem is hard, we would like to compute an approximate solution (to the optimization problem).

What they do is that they show how to find a tree $T$ with $\Delta(T)$ being at most $2\text{OPT} + \log_2 n$. The intuition is that the local search tries to make local improvements to a candidate tree, and when it can't make any improvements, it has found a locally optimal tree. It can then be shown that for any locally optimal tree, the maximum degree is at most $2\text{OPT}+\log_2n$, where $n$ is the number of vertices. The explanation for the logarithmic term is given in Theorem 2.19; it results from a tight analysis.

The best we can hope to do assuming $\sf P \neq NP$ is to find a tree $T$ with $\Delta(T)$ being at most $\text{OPT}+1$. This in fact possible to do, and is shown in a later section of the book.

StackExchange Computer Science Q#23139, answer score: 5