[I]t is difficult to narrow down an exact definition of a tree because every tree is different. So to define a tree, we are actually defining everything else as not a tree, until we decide it fits into the tree category.
Welcome to SE Philosophy! This is a very philosophical question, and one that hinges on the nature of definition. In your title you use the phrase "true definition", and I think I speak for many who would say that there is no such thing as a "true" definition if you mean the phrase to mean universal, beyond argument, so on. Most definitions are used by convention, which is to say there is a general agreement and use, but no authority. You have some options:
Use a soft dictionary like Webster's or the OED. These definitions are generally simple and agreeable to most people. Another place you might find a definition is among biologists, particular botanists who study trees. Believe it or not, not all living organisms fit into the neater categories of biologists devise and they might argue over the definition of an organism.
But this is a question over what a definition really is, and that's where philosophy steps in! For instance, definitions, broadly speaking are intensional and extensional. What you are doing is trying to capture the essence of the tree by speaking to what it means (semantics) to be a tree. Another way would be to point to all trees or list all of the particular members of the set of trees, which is considered extensional. Having so many different kinds of definitions might be confusing, and there are more types, like the operational definition!
So, to answer your question specifically, to some extent, people must agree to a definition of a tree before attempting to use the term. To complicate matters, the famous philosopher Ludwig Wittgenstein anticipated findings in cognitive science which show that sometimes our brains define members of a category even if they have no properties in common at all! Wittgenstein noticed a category like "game" has members who share "family resemblances" but have no properties exactly in common. This has been validated by the linguist Elanor Rosch who showed our brains build prototypes to categorize concepts.
A related term in philosophy is that of natural kinds whose own definition is controversial among philosophers showing that not only do philosophers disagree over definitions, but disagree over the definition of definitions!
Notice on the comments of your question, the question of what a "tree" is depends on whether you're a biologist and study those trees or a computer scientist studying trees. Why use both? Because our brains work with conceptual metaphors.
Like most questions in philosophy, your question can only be satisfied, and not necessarily answered with absolute authority. That's why questions like yours tend to just raise more questions! (What does 'is' mean when one says "That is a tree"? Or how do we know for sure that we agree when we claim to be a tree is truly a tree?