Well if we break Wind down to it's basic components, it is an air pressure differential. High pressure flowing into low pressure.
We can use the 'small-scale' sea breezes as an example. On sunny days the sun rays warm up the land quicker than it can warm up the sea (which has a greater capacity to absorb and store the sun's rays). This results in a temperature difference.
The warm land then heats up the air above it. As the air becomes less dense than the cooler air over the sea, it begins to rise. The rising air over the land then results in lower air pressure over the land. The cooler sea air remains denser and therefore has a higher pressure over the sea. This creates the pressure differential. Nature tries to equalize this difference, and the sea breeze is born. Air flows from the cooler sea into the warmer land, giving some much appreciated relief.
To use this for your WindTree, you would need to provide some mechanism for the WindTree to absorb and store alot more of the sun's rays than the land around it. Your WindTree will act like the sea in this example.
This wind could possibly be 24 hours using katabatic and anabatic wind processes. During the day, the air around the WindTree would be cooler than the surrounding land and wind would blow away from it. During the night, the air around the WindTree could be warmer than the surrounding land and wind would blow towards it.
I do not know how strong you could make this wind or if the tree could 'choose' when to release stronger wind. I do think you could have stronger winds just after the peak period of temperature difference than during periods of smaller temperature differences.
I will leave you to work out a mechanism for the WindTree to absorb more heat without frying. Remember there is a difference between absorb (stored, leaving the air cooler) and reflect (immediately back in the air and hence warmer). You could help the WindTree out by adding more reflective surface features to the land around the WindTree. Eg grass/plant pigments or artificial surfaces etc. This will allow you to handwave less.
I'm not too caught up on my Tree physiology. I am assuming the tree would need major internal changes to handle all this extra heat without it escaping back out into the surrounding air.
Currently I have been thinking of a very extensive root network allowing for more heat to be stored in the tree. The WindTree could then also release or redirect warmth back into the surrounding land through the furtherest roots leaving the air directly around the WindTree itself cooler. Coupled with some sort of ultra absorbant tree surface. I don't think the WindTree will have a very wide surface area, as I think that would allow for more heat to be dissipated back into the air.
Why would it need to do this? Not sure. The WindTree would need some major benefit from storing all this heat and not just releasing it back into the air.
Possibly the plant mechanism involved was evolved during a really cold period in the planet's past. Something to do with needing warmth to create new growth in arctic cold conditions? Could be it needed to defrost the frozen ground around it and hence absorbed more heat from the surrounding air to spread along the root network.
Possibly needed to constantly move its branches to avoid a parasite from settling on it. So it needed a constant breeze? Or to attract a symbiotic creature that is attracted by cooler temps?