Is the monster Connors grandfather?...
I know this is far fetched, so bare with me on this.
When I saw the picture of Connors mum as a kid and noticed her dad was played by Liam Neeson. That made me question why the monster visited Connor in his dreams.
Connors mum said to him "I wish you could have met him", referring to her dad. Obviously, the tree is voiced by Liam, linking the two characters.
Connors dad left him when he was young and he didn't get on with his grandmother, so he could only turn to his grandfather, who he has heard great things about. But the only way of doing this is visualizing him in his dreams.
Another link, is the drawings in Connors mums bedroom that match the stories the tree told Connor in his dreams. These stories could have been told to his mum, by his grandfather and his mum told him when he was younger. Also, his mum mentioned a few times, that the tree was their "friend". She also "saw" the tree behind Connor before she sadly died, giving the tree the look of thanks.
Just an interesting idea I had. Wanted to share, to see what input people can give.