We want to do a preorder traversal over all the nodes in the tree.

preorder(root)

1. Recursion

The recursion for a preorder traversal is to visit the current node, then all the nodes to the left, then right, c L R.

When we visit each node, we add its value to an array. Here's the recursion:

values = []
def preorder(node):
    values.append(node.val) # add value to array
    preorder(node.left)     # visit left children
    preorder(node.right)    # visit right children

2. Base case

This recursion will keep calling itself on deeper and deeper nodes in the tree. Eventually, it calls itself on a null node, preorder(null). In this case, we don't want to traverse any further and we should just do nothing.

preorder(null) = 'do nothing'

3. Code

To code the preorder traversal function, we just combine the recursion and base case. After calling this function, we return the values array that it filled in.

def preorderArray(root):

    values = []
    
    def preorder(node):
        # base case
        if node is None:
            return
        
        # recursion
        values.append(node.val)
        preorder(node.left)
        preorder(node.right)

    preorder(root) # fills in `values` array

    return values

Time Complexity $O(n)$. We walk through all of the nodes in the tree, which is $n$ nodes total.

Space Complexity $O(n)$. The recursive function calls are stored inside of the Call Stack, which grows up to the depth of the tree. The tree might be a flat line, in which case it's $n$ nodes deep.