Divide and conquer algorithm and incremental algorithm are types of algorithms that usually come with recursive functions. In this sense, There are several general tips I may offer:

1. Get clear about what you have got. You can do this by describing your function in a mathematical way, such as using notations like $$$f(x,y)$$$. For example, in the well-known algorithm, mergesort, you can write the function down like $$$\text{MergeSort}(A)$$$, where the argument refers to a sequence $$$A$$$ you have, which you are going to sort.

2. Get clear about what you want your function to put out. In the mergesort case, when you call $$$\text{MergeSort}(A)$$$, you want to sort the elements in $$$A$$$. So, the output of this function should be the sorted $$$A$$$. This is especially essential because it can help you to get rid of imagining large recursive tree in your mind -- you can simply skip it by knowing what the outcome will exactly be.

3. Think over what you should do in one step. Either a D&C algorithm or a incremental algorithm focuses on separating a complex problem into several small and easy-to-handle steps, so it's important to get clear about these steps. Take mergesort as an example again. Since it's a D&C algorithm, so when we are dealing with $$$\text{MergeSort}(A)$$$, we must think about:

• What should we do to divide the problem into small parts? In this case, it's easy. let $$$n$$$ be the number of elements in $$$A$$$, and number the elements in $$$A$$$ from $$$1$$$. We just need to take the middle point $$$m=\lfloor\frac{n+1}{2}\rfloor$$$, and the sequence is naturally divided into two parts, that is $$$A_L=[A_1,A_1,\dots,A_{m}]$$$ and $$$A_R=[A_{m+1},A_{m+2},\dots,A_{n}]$$$.

• What will happen if we call the function recursively? In this case, we will call function $$$\text{MergeSort}(A_L)$$$ and $$$\text{MergeSort}(A_R)$$$ to get $$$A'_L=\text{MergeSort}(A_L)$$$ and $$$A'_R=\text{MergeSort}(A_R)$$$ respectively. As we have already known, $$$A'_L$$$ and $$$A'_R$$$ are both sorted sequences.

• What should we do to conquer the result of the sub-problems? Here, we've got $$$A'_L$$$ and $$$A'_R$$$, two sorted halves of original $$$A$$$, so the aim of conquering part is to merge two sorted sequences into one sorted sequence. Details of merging are skipped.