With the question of a thing's self-identity (which presents no difficulty) the Buddha's Teaching of anattā has nothing whatsoever to do

You say 'The word flux means continuous change. If this idea is applied to everything it would be correct to say that what I see now, e.g. a tree, is not the same as I continue to watch it as it is subject to continuous change' and also 'I have heard as an extension of the same idea, Buddhist monks saying, pointing to an object, that the object is not there'. This doctrine is a complete misunderstanding and is wholly misleading. And, as you quite rightly point out, it is based on the notion of universal flux. In order, therefore, to undermine this false doctrine, it is necessary to point out that the notion of flux, at least as applied to experience, is a self-contradiction.

But why, if it is false, is this doctrine taught? The answer is, because it provides a conveniently simple interpretation of the Suttas, easily learned and easily preached. The Buddha has said that 'What is impermanent, that is suffering; what is suffering, that is not-self'. This is understood (or rather, misunderstood) in the following way.

Impermanence is taken to mean continuous change (flux), and (as you have said) if this notion is correct, the idea of a thing's continuing self-identity cannot be maintained—what appears to be the self-same tree persisting in time is not really the same since it is continuously changing. In consequence of this, the idea of self is an illusion; and it only persists on account of our avijjā, or ignorance of the truth of universal flux. If we remove this ignorance, we shall see that what we formerly took to be a lasting (or existing) selfsame tree ('A = A', the Principle of Self-identity) really has no abiding self at all—it does not really exist. And this explains why 'what is impermanent, that is not-self'. And what is wrong with this? What is wrong with it is—as perhaps you have noticed—that it does not explain why what is impermanent is suffering, and what is suffering is not self.

Suffering (dukkha) is the key to the whole of the Buddha's Teaching,[c] and any interpretation that leaves suffering out of account (or adds it, perhaps, only as an afterthought) is at once suspect. The point is, that suffering has nothing to do with a tree's self-identity (or supposed lack of self-identity): what it does have to do with is my 'self' as subject (I, ego), which is quite another matter (see PARAMATTHA SACCA §6). As I point out (ATTĀ), 'With the question of a thing's self-identity (which presents no difficulty) the Buddha's Teaching of anattā has nothing whatsoever to do: anattā is purely concerned with "self" as subject'. But this is very much more difficult to grasp than the misinterpretation based on the notion of flux, so flux inevitably gets the popular vote (like the doctrine of paramattha sacca, of which it is really a part). The misinterpretation is actually of Mahāyānist origin; and in one of their texts (Prajñāpāramitā) it is specifically stated that it is only on account of avijjā that things appear to exist, whereas in reality nothing exists. But the fact is that, even when one becomes arahat, a tree continues to have a self-identity; that is to say, it continues to exist as the same tree (though undergoing subordinate changes on more particular levels—falling of leaves, growth of flowers and fruit, etc.) until it dies or is cut down. But for the arahat the tree is no longer 'my tree' since all notions of 'I' and 'mine' have ceased.

[c] 'Both formerly, monks, and now, it is just suffering that I make known and the cessation of suffering.' M. 22

Nanavira Thera