Is it real I heve to cut down the trees when the moon is going down?
In case of affermative answer, what is the why?

Yes, there is a connection.

When the moon is full, there is more sap in the wood. This may be
better or worse for building, depending on opinion...

Two quotes, for example:

"Workers who followed traditional methods cut the mesquite tree posts,
center beams, and the rest of the wood used for making the jacales
during a full moon. They believed any part of the tree cut during the
new crescent-shaped moon retained the sap and so attracted insects,
which left a powdery mess that rotted the wood. Houses made of such
wood would not last long."
http://www.texancultures.utsa.edu/ra...chapter011.htm

"Napoleon I. directed that the felling of naval timber should take
place only from November 1 to March 15, and during the decrease of the
moon, on account of the rapid decay of timber, through the
fermentation of its sap, if cut at other seasons."
http://72.1911encyclopedia.org/D/DR/DRY_ROT.htm

Hope this helps.

regards
Marcus