Tiger Tree Hash vs generic Merkle Tree Is there any advantage of using Tiger Tree Hash over any other hash function organized as the Merkle tree? Are there maybe any properties of TIGER that, say, SHA-2 or BLAKE in Merkle tree do not have?

And in general, is there any point of chosing Tiger Tree Hash over anything else in modern applications for file sharing purposes?
The Tiger hash function was mainly a notable thing back in the days as it was designed to make efficient use of 64 bit processors. 

Besides some other uses, Tiger was (and sometimes still is) often used for file integrity verification purposes in file sharing proggies, by using the Tiger hash function to build a Merkle tree… resulting in a so called “Tiger Tree Hash”. (The main advantage of such a tree is that you can verify the integrity of each subtree, even when the data is not yet completely available… as it happens – for example – when downloading torrent files).

Personally, I don't see many advantages to SHA-2 or BLAKE. Especially, since Blake etc. were also build with an eye on efficiency when it comes to 64 bit processors – which are somewhat standard for desktop computers et al nowadays.

As CodesInChaos already indicated, the only reason to use Tiger at all would be compatibility with existing protocols. For new protocol designs, I personally would skip looking at Tiger.