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I started with this aproach too, however this only checks that the children nodes are correct for the parent node only.
Consider the case where your root node is 100, then we traverse down the left side a couple of times and come to a node that is 50. Let's say the left child is 10, and the right is 999. This will pass because the code only checks the immediate children, however it is not a BST because 999 is much bigger than the root node, 100.