#### Study site

Our study site was Creag Meagaidh National Nature Reserve, which occupies 3 940 ha within the central Scottish Highlands (56°57′N, 4°35′W). Mean annual precipitation is approximately 1 250 mm and the major soil type is humic podzols. Vegetation cover is primarily moorland dominated by heather (*Calluna vulgaris*) and purple moor grass (*Molinia caerulea*), with upland woodland occurring in small isolated patches and consisting of primarily downy birch (*B. pubescens*), and to a lesser degree silver birch (*B*. *pendula*), rowan (*Sorbus aucuparia*), and willow (*Salix* spp.). Red deer have been killed since 1986 in order to encourage woodland and scrub regeneration, reducing deer population densities from approximately 17.5 deer km^{−2} in 1986 to 1.7 deer km^{−2} by 2008, based on late winter population counts (Putman et al. 2005). Sheep (*Ovis aries*), roe deer (*Capreolus capreolus*), and sika deer (*Cervus nippon*) also occur in the region, but in very low numbers.

#### Developing a model of birch invasion in the Scottish Highlands

Our approach for predicting birch population dynamics is entirely based on fitting mathematical functions at the levels of size-classes and individuals to field data collected at Craeg Mageidh (e.g., Pacala et al. 1996; Staver et al. 2009). The simulation model starts with a map of adult tree locations and makes spatially explicit predictions of how birch woodland spreads over time (Fig. 1). At each time step, the model uses the current distribution of adult trees (>3 m tall) to predict the number of seedlings recruited within every square meter of the landscape. Growth and survival of seedlings of different heights are then predicted by a series of probabilities that determine the proportion of seedlings surviving between years and moving from a small (0–2 m) to large (2–3 m) height class. These transition probabilities are ultimately influenced by deer browsing. Once seedlings grow beyond 3 m tall, they are classified as seed-bearing adults. Finally, the model grows and kills the adults. This routine is followed from the first to last year of the simulation (Fig. 1; see User Manual in Appendix S1 for description targeted for nonspecialists and conservation practitioners).

The simulation model was characterized by three “submodels”: juvenile recruitment, juvenile growth and mortality, and adult growth and mortality. Full details for data collection and parameter estimation are given in supporting information, so we summarize them here. First, we measured juvenile recruitment by surveying birch seedlings (<2 m height) and adult trees (≥3 m height) in six plots ranging from 0.6–4.0 ha. We recorded the height and location of each adult within a 300-m radius of the plots, and within 160–427 quadrats per plot (0.5 × 0.5 m in size), we counted the number of seedlings and visually estimated the ground cover of *Agrostis*-*Festuca* grassland, bog myrtle (*Myrica gale*), bracken fern (*Pteridium aquilinum*), heath (*Erica* spp.), heather, moss, purple moor grass, and *Vaccinium* spp. We assumed that seed dispersal from adults reached a peak within several meters and then declined exponentially. This was best described by a log-normal function and we fitted this function to our observations using maximum-likelihood methods (Greene et al. 2004). Model selection techniques showed that the log-normal model was more strongly supported than other functions, including those that allowed dispersal to be directionally dependent (Science Manual, Appendix S2). Our data also enabled us to estimate the potential number of recruits produced by each adult (*STR*) and favorability of ground cover for seedling establishment. We did so by simultaneously predicting the shape of the dispersal function, favorability of different ground cover, and value of *STR* that maximized the fit between predicted and observed seedling counts, given the observed spatial locations of adult trees. The different substrates also implicitly allow us to consider how establishment varies with different levels of understorey light and litterfall (e.g., *C. vulgaris* vs. *Vaccinium myrtillus*, Hester et al. 1991).

Second, we estimated juvenile growth and mortality using an 8 year dataset monitoring birch trees along six 1-km long transects spanning a gradient of 0–100% of trees being browsed. At 100 m intervals along each transect, a 100 × 2-m plot was located in which all trees emerging from unique seedlings were counted in 0–2, 2–3, and >3 m size-classes. The number of leader stems visibly damaged by mammalian herbivores was also recorded. We then estimated transition probabilities among the three height classes, and how these were affected by observed levels of browse damage, within a hierarchical Bayesian framework using Markov chain Monte Carlo sampling. This approach also allowed us to estimate the degree of interannual and spatial variation in growth and mortality. Matrix models such as ours have been successfully used to predict the effects of herbivores on woodland regeneration over medium-term timescales (e.g., 200 years, Staver et al. 2009), despite being criticized as inferior to approaches that incorporate individual-level variation, for example, integral projection models (IPMs, Ramula et al. 2009). We chose not to apply IPMs to our data as it would have required tenuously assuming how individual variation might be expressed among juvenile trees, and would have produced similar results to matrix models given that our dataset was suitably large (Ramula et al. 2009), and that growth in taller height tiers was neither size dependent nor autocorrelated (Pfister and Stevens 2003).

Third, we extracted increment cores from 40 adult birch trees at Corrour Estate, directly south of Craeg Mageidh. For each tree, we also recorded the diameter at breast height, standing height, and crown area. We calculated annual radial growth, averaged over the previous 8 years, and used this to predict adult height growth from standard allometric relationships (e.g., Russo et al. 2007). We then predicted the population size structure of all adults across our two sites using the observed growth rate but estimating density-independent mortality (Coomes et al. 2003). Adult mortality thus corresponded with the value that maximized the fit between the observed and the predicted size distributions. We did not simulate density-dependent mortality arising from competition for light as our interest was in tracking birch invasion rather than the dynamics of established stands. Density dependence is likely to also be weak for *Betula* as it regenerates poorly beneath canopy cover, often developing as nearly even-sized stands (Atkinson 1992; Mountford and Peterken 2000). However, we did remove adult trees from simulations when their crown area was 90% overtopped by neighbors. We also limited the number of individuals that could be recruited into 1 m × 1 m units of our landscape, whereby there was a maximum of 50 juveniles of any age <3 m tall in each 1 m^{2} area.

#### Model validation

We validated our model by comparing predicted numbers of juvenile trees (<3 m tall) with observed counts in 2 × 100 m plots, accounting for the initial ground cover and annual variation in deer browsing in each plot. Counts were recorded in 10 plots located along each of five permanent transects in 2000. These data preceded measurements used to parameterize juvenile growth and mortality, so validation was independent of model parameterization. Starting conditions for validations were set by identifying all adult trees within 300 m of transects using digitized color aerial photographs from May 1990. We then parameterized our simulation model with adult tree distributions in the first year that a transect was measured, corresponding with 1988 for three transects and 1992 for the remaining two, and ran the model until 2000. The median count for each plot was calculated from 1000 simulations rather than the mean because distributions of predicted counts had very long right tails. We fitted a model to predict the median values from observed counts using a generalized linear model with Poisson error structure. The standard errors of the model parameter estimates were corrected by estimating a dispersion parameter to account for over dispersion (Ver Hoef and Boveng 2007). To test whether the slope and intercept of the model overlapped 1 and 0, respectively, as expected for an unbiased relationship between predicted and observed values, we calculated 95% confidence intervals (CIs). All models were fitted in *R* v2.14 (R Development Core Team 2011).

We also varied the most important predictors of juvenile tree densities: the potential number of recruits produced by each adult (*STR*) and survival in the 0–2 m height tier (*s*_{1}), by 80–110% to try and minimize any bias in the predicted counts. We then selected the values of *STR* and *s*_{1} that produced a validation slope and intercept closest to 1 and 0, respectively, and used these in subsequent simulations. Our validation assumed: (1) ground cover varied little over time, so 2009 surveys could predict substrate favorability, (2) there was no competition between trees established along transects, if any, and incoming recruits, (3) soil seed banks were negligible (Miller and Cummins 2003), and (4) differences between predicted and observed values accumulate minimally over time (see Appendix S3 for further discussion).

#### Statistical analyses

We tested how birch regeneration was relatively influenced by deer browsing, substrate favorability, seed availability, and active management, corresponding with the planting of small patches of adults (“patch invasion model”). Our approach was to fit generalized linear models to predict juvenile tree densities and adult BAs from each level of simulations given our four aforementioned variables. We modeled juvenile tree counts in 0–2 and 2–3 m height classes across our entire landscape using a Poisson distribution with log-link function and accounting for over dispersion as in the validation. By contrast, adult BA was log-transformed and modeled over the same spatial extent using a Gaussian error structure. Active management was simply included as a binary predictor (0 = core invasion; 1 = patch invasion). We scaled our four explanatory variables to a mean of zero and SD of one so that their effects and 95% CIs would be directly comparable. All analyses were performed using *R*.