In most cases I see authors write: oak tree, pine tree, palm tree, and etc. But can we just call them an oak, a pine, a palm?
- The house was standing next to an oak (instead of an oak tree).
- I was climbing a pine (instead of a pine tree).
You can, but many trees bear a fruit or other accessory that is known by the same name: a cherry, an apple, a walnut. Even in your examples, a "palm" is often a palm frond, and a "pine" is a needle of a pine tree.
Question: In most cases I see authors write: oak tree, pine tree, palm tree, and etc. But can we just call them an oak, a pine, a palm?
Response: for some trees yes, you do not need to say tree: A pine, an oak.
For others you do need tree: I was climbing a walnut tree.
If the word for the type of tree refers to the type of wood that characterizes the tree, you don't need the word tree. Mostly you can just use the noun. But not for: fruit-bearing trees. And that includes walnuts.
Because walnut is a type of wood and the fruit of a tree. Also, cherry, there is cherry wood and cherries that grow on trees. So, I was climbing a cherry tree. Not: I was climbing cherries.
If the context is clear, you can say palm for palm tree even though a palm is not a type of wood.
I was climbing the jacaranda or bougainvillea in the yard. [those words are only the type of wood, not a fruit of the tree].