Birds eat bugs, and bugs eat plants. This is called a trophic cascade. So it might follow that if there were fewer birds, there would be more bugs and more damage to plants. Scientists, however, have had trouble documenting this. Some studies have reported more leaf damage, and some studies have not.
A study at Hubbard Brook in the New Hampshire mountains set out to exhaustively monitor the trophic cascade by setting up study plots at low, mid, and high elevations and over multiple years (2004 to 2006). The plant they studied is a large shrub called the striped maple (Acer pensylanicum).
In some study plots the striped maple was fenced in so birds could not eat the bugs inside. In other plots researchers removed bugs from the maples. A third set of maples were left as they were.
The bugs in the study are mostly caterpillars, prime food for birds to feed their young. Below are pictures of the most common birds observed foraging in the study plots.

Blue-headed Vireo

Red-eyed Vireo

Black-capped Chickadee

Winter Wren

Veery

Swainson's Thrush

Hermit Thrush

Nashville Warbler

Magnolia Warbler

Black-throated Blue Warbler

Yellow-rumped Warbler

Black-throated Green Warbler

American Redstart

Ovenbird

Rose-breasted Grosbeak

Scarlet Tanager

Dark-eyed Junco

Canada Warbler
The study did find that there were differences between the years. It found that in 2004 there was substantially more damage to leaves due to a caterpillar outbreak. There were also differences in the elevation; the highest elevation plants grew more, presumably because of more light (due to a shorter tree canopy and more gaps in the canopy).
But keeping birds away from the plants did not result in more bugs on the plants. It seems that the bugs are so well camouflaged that the birds have trouble finding them.
For the plants, only about 5% of the leaves were damaged by insects, regardless of the plant's location. Removing bugs from plants did not result in the plant growing more. Plant growth seems to be governed more by the environment and climate than by insect's eating them.
This article summarizes the information in this publication:
Schwenk, W. S., Strong, Allan M. and Sillett, Terence Scott. Effects of bird predation on arthropod abundance and tree growth across an elevational gradient, Journal of Avian Biology, 41 (4) 367-377. 2010.
Considerable uncertainty surrounds the conditions under which birds can cause trophic cascades. In a three-year experiment, we studied the direct and indirect effects of insectivorous birds on arthropod abundance, herbivory, and growth of striped maple Acer pensylvanicum saplings in a northern hardwood forest of central New Hampshire, USA. We manipulated bird predation by erecting exclosures around saplings and directly manipulated herbivory by removing herbivores. We also examined how climate modifies these interactions by replicating the experiment at three locations along an elevational gradient. Effects of bird predation were variable. Overall, mean arthropod biomass was 20% greater on saplings within bird exclosures than on controls (p<0.05). The mean biomass of leaf-chewing herbivores, primarily Lepidoptera larvae, was 25% greater within exclosures but not statistically different from controls. To a lesser degree, mean herbivore damage to foliage within exclosures exceeded that of controls but differences were not significant. We also did not detect significant treatment effects on sapling shoot growth. The high understory vegetation density relative to bird abundance, and low rate of herbivory during the study (mean 5% leaf area removed, controls), may have limited the ability of birds to affect sapling growth. Climate effects operated at multiple scales, resulting in a complex interplay of interactions within the food web. Regional synchrony of climatic conditions resulted in annual fluctuations in herbivore abundance and tree growth that were shared across elevations. At the same time, local environmental variation resulted in site differences in the plant, herbivore, and bird communities. These patterns resulted in a mosaic of top-down strengths across time and space, suggesting an overall pattern of limited effects of birds on plant growth, possibly interspersed with hotspots of trophic cascades.
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