When we look at tree data structures, there are three important things to understand: time complexity, space complexity, and structural properties. Knowing these metrics helps us make smart choices about which data structure to use for different tasks.

1. Time Complexity
Time complexity shows how long operations take, like adding, removing, or finding a value in a tree. This time can change depending on the type of tree.

For example, in a balanced binary search tree (BST), these operations usually take about $O(\log n)$ time. This means that each time we make a decision, it cuts the options in half.

But in an unbalanced tree, the worst case could take $O(n)$ time. This means it could be really slow. Imagine you’re inserting items into a BST—if the tree is balanced, it’s quick. But if it leans too much, like a crooked line, adding new items takes a lot longer.

2. Space Complexity
Space complexity tells us how much memory we need for a tree. This usually depends on how many nodes (or points) there are in the tree.

A typical binary tree needs space equal to $O(n)$, where $n$ is the number of nodes. Each node has two links (or pointers) along with its data, which uses more memory than simpler structures like arrays that can use space more efficiently.

Special trees, like AVL or Red-Black Trees, need a bit more space because they also hold extra information, like height or color. But this extra information helps keep the tree balanced.

3. Structural Properties
Trees have special features that are important for their performance. For example, the height of a tree (the longest path from the starting point, called the root, to a leaf) affects the time it takes to do operations in the tree.

In well-balanced trees, the height is smaller, making everything work efficiently. But keeping a tree balanced can be tricky, especially when adding or removing nodes. Sometimes we need to rotate or adjust parts of the tree to keep it in shape.

In summary, understanding these complexity metrics is really important for using tree data structures better in real life. Knowing how they work helps us pick the right data structure, so our operations stay quick and efficient.