Trie data structure in C++ with add and search
This is a pretty good implementation from what I see. It's easy to read and looks very straightforward. Nice work!
Naming
For the most part your naming is good. There are a few small things I would change. For example, the get_leaf() method. When I read the name I thought it was going to retrieve a leaf node, but I couldn't see how it could determine which one. In reality, it returns a bool that tells whether it is a leaf or not. I would rename it to is_leaf() to avoid confusion.
I also don't like the word value as it's fairly meaningless. It represents a letter in the set of words formed by that branch of the trie. I would name it character or letter.
Organization
For the default constructor for a Node, in C++11 and later, you don't need to write it out by hand. In the header, you can just declare it as:
Node() = default;

In trie.cpp you have the implementation of 3 classes. I usually break them out into a separate source file per class. Given how small these are, it's not a huge deal to put it all in one file. I would, however, keep all methods for a class together. You have several methods of Node, then 2 methods of NodePointerComp, then more methods of Node. It makes it a little hard to find things when examining the source. So I'd move all methods from a single class together into 1 section. You might also want to add either a comment break between them (just something like //----NodePointerComp methods----) or, if your compiler supports it, something like a #pragma mark NodePointerComp. In my IDE, when the editor sees #pragma mark &lt;Name&gt;, it adds &lt;Name&gt; to the pop-up menu with all the functions in it. That makes it really easy to navigate to those methods quickly.
Simplify
It looks like you're using the convention of a negative value representing whether a node is the end of a word or not. I originally thought that you were trying to define the last node in a branch, but I see from the comments I was mistaken. That's a good indication that the name is too confusing and the implementation too complex.
I'm not an expert on the C++ spec, but I thought that whether char was signed or not (when not explicitly specified) was implementation-dependent. Even if it's not, that's a tenuous way to determine the end of a word. I recommend a bool that says it's the end of a word. It would simplify the get_leaf() and set_leaf methods(). Also they should be renamed to is_word_end() and set_word_end().
Encapsulation
I'm a little uncomfortable with Node::children being public. That's an implementation detail, and I think it would be a good idea to hide that detail from users of the class. It would make the logic in Trie slightly less straightforward, but I think the tradeoff would be worth it in the long run. Right now, Trie can modify that variable and Node won't necessarily be aware of it. It also makes it more difficult to use in a multithreaded environment, should you need to do that. If it were private, you could ensure that it was never accessed outside of some sort of lock (or mutex, whatever) inside the class.