With a dew pond there is no spring involved. It is a perched (non-water table) pond that derives its water mostly from condensate. To get condensate, you need an open sky to which heat can be radiated at night so the temperature of the insulated (from geothermal heat) pond bottom can cool to temperatures below the dew point and start condensing moisture. A tree
canopy over the pond would hold in the heat and prevent dew from accumulating (except for any dew forming and dripping off the leaves, a less efficient process of filling the pond since the tree adsorbs a lot of that moisture). Having transpiring vegetation nearby to raise the local relative humidity is good, but it can't over shadow the dewpond for the reason mentioned above.
The whole reason for making a dew pond is so you can have a non-water table water source that isn't dependent on a watershed to
feed rain water to it. It can generate water even in the absence of precipitation as long as the relative humidity is high
enough that an insulated surface exposed to a clear sky can drop to temperatures below the dew point. You can place a dew pond up on the top of a ridge and have a self-refilling, self-regulating water source without the need for pumps or other fancy hardware.
The main maintenance required by a dew pond is to maintain the waterproof layer (keep tree roots and heavy livestock from puncturing it) and the insulating layer underneath it (in some climates, rot and termites can degrade the hay often used to create the insulating layer and you might need to use some non-decaying alternative). The sky above the dew pond needs to remain open of trees so it radiate heat efficiently at night. The only time you would want to maintain some trees overhanging the dew pond was if the pond was located up on a ridge that was often enveloped in low clouds where the fog drip generated by the trees more than made up for the loss of open sky condensed dew. Especially in locations that don't get very many clear nights. Also if too much muck accumulates on the insulated pond floor over time, this added material increases the thermal mass of the insulated layer, slowing down its rate of temperature drop and the amount of time at night that its temperature is below dew point. So, depending on the biological activity of organisms living in the dew pond, its bottom might need to cleared of muck every few decades to main its efficiency. Ideally the insulated surface of a dew pond should be black so it radiates heat better at night.