I've been thinking and a question has arisen. Does this type of compiler really need the complete syntax tree in memory?
Does a Recursive Predictive Descent Parser NEEDS to construct a complete syntax tree and store in memory?
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The parser doesn't need to store anything. It could just tell you whether or not the input is valid. But usually, you are parsing an input for some purpose, which will determine what information (other than simple validity) you need to extract.
For example, a compiler needs to convert the input program into some executable format. It's possible to convert directly to machine instructions (or to some intermediate representation), particularly if you don't care much about optimising the generated code.
Without knowing the purpose of parsing the input, it's impossible to say anything more concrete.